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{SECT 0 {EXCHG {PARA 208 "" 0 "" {TEXT 218 161 "Study of the Second E
lement of the Painlev\351 2 hierarchy using various Darboux coordinate
s and making the link explicit connections with isospectral coordinate
s. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 362 "Loading the Lax pair obt
ained for the second element of the Painlev\351 2 hierarchy in \"Hamil
tonian representation of isomonodromic deformations of general rationa
l connections on $\\mathfrak\{gl\}_2(\\mathbb\{C\})$\". In our case we
 have \\tau_\{\\infty,i\}=2t_\{\\infty,i\}. The auxiliary matrices thu
s also get an extra $2$ factor because of this change of time normaliz
ation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }
{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 12 "tau2:=2*t2;
\n" }{MPLTEXT 1 0 12 "tau1:=2*t1;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 
0 20 "tdL:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 99 "tdL[1,1]:=-lambda^3+((p
1-p2)/(q1-q2)+q1^2+q2*q1+q2^2)*lambda+(-p1*q2+p2*q1)/(q1-q2)-q1*q2*(q1
+q2);\n" }{MPLTEXT 1 0 35 "tdL[1,2]:=(lambda-q1)*(lambda-q2):\n" }
{MPLTEXT 1 0 281 "tdL[2,1]:= (tau2+(2*(p1-p2))/(q1-q2)+2*q1^2+2*q2*q1+
2*q2^2)*lambda^2+((q1+q2)*tau2+tau1+(2*(p1*q1-p2*q2))/(q1-q2)+(2*(q1+q
2))*(q1^2+q2^2))*lambda+(q1+q2)*tau1+(1/4)*tau2^2+(q1^2+q1*q2+q2^2)*ta
u2-(p1-p2)^2/(q1-q2)^2-(2*(p1*q2-p2*q1))*(q1+q2)/(q1-q2)+q1^4+q2^4-q1^
2*q2^2+2*tinfty10;\n" }{MPLTEXT 1 0 21 "tdL[2,2]:=-tdL[1,1]:\n" }
{MPLTEXT 1 0 4 "tdL:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 150 "tdA1:=2*Matrix(2, 2, [[-(1/2)*q1-(1/2)*q2-(1/2)*lamb
da, 1/2], [(tau2*(q1-q2)+2*q1^3-2*q2^3+2*p1-2*p2)/(2*q1-2*q2), (1/2)*q
1+(1/2)*q2+(1/2)*lambda]]);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "t
dA2:=2*" }{MPLTEXT 1 0 343 "Matrix(2, 2, [[(-q2^3-q1*q2^2+(lambda^2+q1
^2)*q2+q1^3-lambda^2*q1+p1-p2)/(4*q1-4*q2), (1/4)*lambda-(1/4)*q1-(1/4
)*q2], [(2*q1^4+2*lambda*q1^3+q1^2*tau2+(lambda*tau2+2*p1+tau1)*q1-2*q
2^4-2*lambda*q2^3-q2^2*tau2+(-lambda*tau2-2*p2-tau1)*q2+2*(p1-p2)*lamb
da)/(4*q1-4*q2), (-q1^3-q1^2*q2+(lambda^2+q2^2)*q1+q2^3-lambda^2*q2-p1
+p2)/(4*q1-4*q2)]]);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 79 "Ham1:=2*unapply( (p1^2-p2^2)/2/(q1-q2)-(1/2)*(q1+q2)*
((q1^2+q2^2)^2-q1^2*q2^2)\n" }{MPLTEXT 1 0 57 "-1/2*(q1^2+q1*q2+q2^2)*
tau1-1/2*(q1+q2)*(q1^2+q2^2)*tau2\n" }{MPLTEXT 1 0 62 "-1/8*(q1+q2)*ta
u2^2 -1/2*(q1+q2)*(2*tinfty10-1), q1,q2,p1,p2);" }{MPLTEXT 1 0 1 "\n" 
}{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "Ham2:=2*" }{MPLTEXT 1 0 8 "unapp
ly(" }{MPLTEXT 1 0 80 "(1/4)*(q1*p2^2-q2*p1^2)/(q1-q2)+1/4*(q1^4+q1^3*
q2+q1^2*q2^2+q1*q2^3+q2^4)*q2*q1\n" }{MPLTEXT 1 0 25 "-(1/4)*1*(p1-p2)
/(q1-q2)\n" }{MPLTEXT 1 0 79 "+(1/4*(q1+q2))*q2*q1*tau1+(1/4*(q1^2+q1*
q2+q2^2))*q2*q1*tau2+1/16*q2*q1*tau2^2\n" }{MPLTEXT 1 0 42 "+(1/4)*q2*
q1*(2*tinfty10-1), q1,q2,p1,p2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 
1 "\n" }{MPLTEXT 1 0 28 "Hinfty1:=Ham1(q1,q2,p1,p2):\n" }{MPLTEXT 1 0 
15 "Hinfty0:=2*Ham2" }{MPLTEXT 1 0 13 "(q1,q2,p1,p2)" }{MPLTEXT 1 0 2 
":\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "omega:=1;\n" }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 17 "dq1dt1:=diff(Ham1" }{MPLTEXT 1 0 13 "(q1,q2
,p1,p2)" }{MPLTEXT 1 0 6 ",p1):\n" }{MPLTEXT 1 0 17 "dq2dt1:=diff(Ham1
" }{MPLTEXT 1 0 13 "(q1,q2,p1,p2)" }{MPLTEXT 1 0 6 ",p2):\n" }{MPLTEXT
 1 0 18 "dp1dt1:=-diff(Ham1" }{MPLTEXT 1 0 13 "(q1,q2,p1,p2)" }
{MPLTEXT 1 0 6 ",q1):\n" }{MPLTEXT 1 0 18 "dp2dt1:=-diff(Ham1" }
{MPLTEXT 1 0 13 "(q1,q2,p1,p2)" }{MPLTEXT 1 0 6 ",q2):\n" }{MPLTEXT 1 
0 17 "dq1dt2:=diff(Ham2" }{MPLTEXT 1 0 13 "(q1,q2,p1,p2)" }{MPLTEXT 1 
0 6 ",p1):\n" }{MPLTEXT 1 0 17 "dq2dt2:=diff(Ham2" }{MPLTEXT 1 0 13 "(
q1,q2,p1,p2)" }{MPLTEXT 1 0 6 ",p2):\n" }{MPLTEXT 1 0 18 "dp1dt2:=-dif
f(Ham2" }{MPLTEXT 1 0 13 "(q1,q2,p1,p2)" }{MPLTEXT 1 0 6 ",q1):\n" }
{MPLTEXT 1 0 18 "dp2dt2:=-diff(Ham2" }{MPLTEXT 1 0 13 "(q1,q2,p1,p2)" 
}{MPLTEXT 1 0 5 ",q2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I%tau2G6\",$*&\"\"#\"\"\"I#t2GF$F(F(" }}{PARA 11 "" 1 "" 
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F'F6F'F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I$tdLG6\"6$\"\"#\"\"\",8
*&,,*&F'F(I#t2GF%F(F(*(F'F(,&I#p1GF%F(I#p2GF%!\"\"F(,&I#q1GF%F(I#q2GF%
F2F2F(*&F'F()F4F'F(F(*(F'F(F5F(F4F(F(*&F'F()F5F'F(F(F()I'lambdaGF%F'F(
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1F(F5F(F2F(F3F2F(*(F'F(F@F(,&*$F7F(F(*$F:F(F(F(F(F(F<F(F(*(F'F(F@F(FBF
(F(*$)F-F'F(F(*(F'F(,(FIF(*&F4F(F5F(F(FJF(F(F-F(F(*&)F/F'F()F3F'F2F2**
F'F(,&*&F0F(F5F(F(*&F1F(F4F(F2F(F@F(F3F2F2*$)F4\"\"%F(F(*$)F5FZF(F(*&F
7F(F:F(F2*&F'F(I)tinfty10GF%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(m
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GFivF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 
"-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6
#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I&mfracGF$6)-F,6%-I#mnGF$6%Q\"2F
'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'-I#moGF$6-Q1&Invisi
bleTimes;F'FC/%&fenceGQ&falseF'/%*separatorGFL/%)stretchyGFL/%*symmetr
icGFL/%(largeopGFL/%.movablelimitsGFL/%'accentGFL/%'lspaceGQ&0.0emF'/%
'rspaceGFen-F#6%-F,60-FG6.Q*&uminus0;F'F@FCFJFMFOFQFSFUFW/FZQ,0.222222
2emF'/FgnF`o-F,6#-I%msupGF$6%-I#miGF$6&Q#q2F'/%'italicGQ%trueF'F@/FDQ'
italicF'-F=6%Q\"3F'F@FC/%1superscriptshiftGQ\"0F'-FG6.Q(&minus;F'F@FCF
JFMFOFQFSFUFWF_oFao-F,6%-Fho6&Q#q1F'F[pF@F^pFF-Feo6%FgoF<Fcp-FG6.Q\"+F
'F@FCFJFMFOFQFSFUFWF_oFao-F,6%-F#6%-F,6%-F,6#-Feo6%-Fho6&Q)&lambda;F'/
F\\pFLF@FCF<FcpF`q-F,6#-Feo6%F[qF<FcpF@FCFFFgoF`q-F,6#-Feo6%F[qF`pFcpF
fp-F,6%F[rFFF[qF`q-Fho6&Q#p1F'F[pF@F^pFfp-Fho6&Q#p2F'F[pF@F^pF@FC-F,6#
-F,6%-F,6%-F=6%Q\"4F'F@FCFFF[qFfp-F,6%FgsFFFgo/%.linethicknessGQ\"1F'/
%+denomalignGQ'centerF'/%)numalignGFat/%)bevelledGFLF@/%)rowalignGQ!F'
/%,columnalignGFht/%+groupalignGFht/%(rowspanGF^t/%+columnspanGF^t-F56
(-F,6'-F,6%-F86)-F=6%F^tF@FCF<F\\tF_tFbtFdtF@FF-F,6#F]rFfp-F,6%FguFF-F
,6#F[qFfp-F,6%FguFF-F,6#FgoFftFitF[uF]uF_uFftFitF[u-F26'-F56(-F86)-F,6
%F<FF-F#6%-F,63-F,6%F<FF-Feo6%F[qFgsFcpF`q-F,6'F<FFF]rFFFgrF`q-F,6'F<F
FFcrFF-Fho6&Q#t2F'F[pF@F^pF`q-F,6%-F#6%-F,6'-F,6'F<FFF]rFFFiwF`q-F,6%F
<FFF[sF`q-F,6%F<FF-Fho6&Q#t1F'F[pF@F^pF@FCFFF[qFfp-F,6%F<FF-Feo6%FgoFg
sFcpFfp-F,6'F<FFF]rFFFdoFfp-F,6'F<FFF^qFFFiwF`q-F,6%-F#6%-F,6(F\\oFbxF
fp-F,6%F<FFF^sFfpFfxF@FCFFFgoF`q-F,6'F<FF-F#6%-F,6%F[sFfpF^sF@FCFFF]rF
@FCFasF\\tF_tFbtFdtF@FftFitF[uF]uF_u-F56(-F86)-F,6%F<FF-F#6%-F,60F\\oF
erFfp-F,6%FcrFFFgoF`q-F,6%-F#6%-F,6%FiqF`q-F,6#F^qF@FCFFF[qF`qFboFfp-F
,6%F[rFFFgoFfpF[sF`qF^sF@FCFasF\\tF_tFbtFdtF@FftFitF[uF]uF_uFftFitF[u/
%&alignGQ%axisF'/FgtQ)baselineF'/FjtFat/F\\uQ'|frleft|hrF'/%/alignment
scopeGF]p/%,columnwidthGQ%autoF'/%&widthGFc\\l/%+rowspacingGQ&1.0exF'/
%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF^]l/%&fra
meGF^]l/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFL/%-equalcolumnsGF
L/%-displaystyleGFL/%%sideGQ&rightF'/%0minlabelspacingGF[]lF@FC/%%open
GQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Ham1G6\",$*
&\"\"#\"\"\"f*6&I#q1GF$I#q2GF$I#p1GF$I#p2GF$F$6$I)operatorGF$I&arrowGF
$F$,.*(#F(F'F(,&*$)F-F'F(F(*$)F.F'F(!\"\"F(,&F+F(F,F:F:F(*(F4F(,&F+F(F
,F(F(,&*$),&*$)F,F'F(F(*$)F+F'F(F(F'F(F(*&FEF(FCF(F:F(F:*&,(FDF(*&F,F(
F+F(F(FBF(F(I#t1GF$F(F:*(F=F(FAF(I#t2GF$F(F:*(F4F(F=F()FLF'F(F:*(F4F(F
=F(,&*&F'F(I)tinfty10GF$F(F(F(F:F(F:F$F$F$F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%Ham2G6\",$*&\"\"#\"\"\"f*6&I#q1GF$I#q2GF$I#p1GF$I#p2GF
$F$6$I)operatorGF$I&arrowGF$F$,0*(#F(\"\"%F(,&*&)F-F'F(F,F(!\"\"*&)F.F
'F(F+F(F(F(,&F+F(F,F9F9F(**F4F(,,*$)F+F5F(F(*&)F+\"\"$F(F,F(F(*&)F+F'F
()F,F'F(F(*&F+F()F,FCF(F(*$)F,F5F(F(F(F,F(F+F(F(*(F4F(,&F-F(F.F9F(F<F9
F9*,#F(F'F(,&F+F(F,F(F(F,F(F+F(I#t1GF$F(F(*,FNF(,(*$FEF(F(*&F,F(F+F(F(
*$FFF(F(F(F,F(F+F(I#t2GF$F(F(**F4F(F,F(F+F()FVF'F(F(**F4F(F,F(F+F(,&*&
F'F(I)tinfty10GF$F(F(F(F9F(F(F$F$F$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I&omegaG6\"\"\"\"" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 218 105 "Fir
st change of Darboux coordiantes: From (q,p) to (Q,P). This change is \+
time-independent and symplectic." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 
118 "We first compute the change of coordinates and the evolutions of \+
the new variables Qinfty0, Qinfty1, Pinfty0, Pinfty1." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "solve(\{p1=omega*Pinfty0*q2-omega*P
infty1,p2=omega*Pinfty0*q1-omega*Pinfty1\},\{Pinfty0,Pinfty1\});" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 59 "q1sol:=(-Qinfty1-sqrt(Qinfty1^2-
4*omega*Qinfty0))/2/omega;\n" }{MPLTEXT 1 0 58 "q2sol:=(-Qinfty1+sqrt(
Qinfty1^2-4*omega*Qinfty0))/2/omega;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 
1 0 7 "p1sol:=" }{MPLTEXT 1 0 32 "omega*Pinfty0*q2-omega*Pinfty1;\n" }
{MPLTEXT 1 0 7 "p2sol:=" }{MPLTEXT 1 0 32 "omega*Pinfty0*q1-omega*Pinf
ty1;\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 34 "Qinfty1-(-omega*
(q1sol+q2sol))); \n" }{MPLTEXT 1 0 39 "simplify(Qinfty0-(omega*q1sol*q
2sol));\n" }{MPLTEXT 1 0 12 "Qinfty1sol:=" }{MPLTEXT 1 0 16 "-omega*(q
1+q2):\n" }{MPLTEXT 1 0 12 "Qinfty0sol:=" }{MPLTEXT 1 0 12 "omega*q1*q
2:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "Pinfty0sol := -(p1-p2)/(q1-
q2):\n" }{MPLTEXT 1 0 38 "Pinfty1sol := -(p1*q1-p2*q2)/(q1-q2):\n" }
{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 40 "p1sol-(omega*Pinfty0*q2-o
mega*Pinfty1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }
{MPLTEXT 1 0 40 "p2sol-(omega*Pinfty0*q1-omega*Pinfty1));" }{MPLTEXT 
1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "dQinfty1dt1inter:=un
apply( simplify( diff(" }{MPLTEXT 1 0 10 "Qinfty1sol" }{MPLTEXT 1 0 
12 ",q1)*dq1dt1+" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 10 "Qinfty1sol
" }{MPLTEXT 1 0 12 ",q2)*dq2dt1\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 
5 "diff(" }{MPLTEXT 1 0 10 "Qinfty1sol" }{MPLTEXT 1 0 12 ",p1)*dp1dt1+
" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 10 "Qinfty1sol" }{MPLTEXT 1 0 
12 ",p2)*dp2dt1+" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 10 "Qinfty1sol
" }{MPLTEXT 1 0 4 ",t1)" }{MPLTEXT 1 0 10 "),p1,p2):\n" }{MPLTEXT 1 0 
91 "dQinfty0dt1inter:=unapply( simplify( diff(Qinfty0sol,q1)*dq1dt1+di
ff(Qinfty0sol,q2)*dq2dt1\n" }{MPLTEXT 1 0 74 "+diff(Qinfty0sol,p1)*dp1
dt1+diff(Qinfty0sol,p2)*dp2dt1+diff(Qinfty0sol,t1)" }{MPLTEXT 1 0 10 "
),p1,p2):\n" }{MPLTEXT 1 0 91 "dPinfty1dt1inter:=unapply( simplify( di
ff(Pinfty1sol,q1)*dq1dt1+diff(Pinfty1sol,q2)*dq2dt1\n" }{MPLTEXT 1 0 
74 "+diff(Pinfty1sol,p1)*dp1dt1+diff(Pinfty1sol,p2)*dp2dt1+diff(Pinfty
1sol,t1)" }{MPLTEXT 1 0 10 "),p1,p2):\n" }{MPLTEXT 1 0 91 "dPinfty0dt1
inter:=unapply( simplify( diff(Pinfty0sol,q1)*dq1dt1+diff(Pinfty0sol,q
2)*dq2dt1\n" }{MPLTEXT 1 0 74 "+diff(Pinfty0sol,p1)*dp1dt1+diff(Pinfty
0sol,p2)*dp2dt1+diff(Pinfty0sol,t1)" }{MPLTEXT 1 0 9 "),p1,p2):" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 37 "dQinfty1dt1
inter2:=unapply(simplify(d" }{MPLTEXT 1 0 15 "Qinfty1dt1inter" }
{MPLTEXT 1 0 23 "(p1sol,p2sol)),q1,q2):\n" }{MPLTEXT 1 0 19 "dQinfty0d
t1inter2:=" }{MPLTEXT 1 0 8 "unapply(" }{MPLTEXT 1 0 25 "simplify(dQin
fty0dt1inter" }{MPLTEXT 1 0 14 "(p1sol,p2sol))" }{MPLTEXT 1 0 7 ",q1,q
2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 19 "dPinfty1dt1inter2:=" }
{MPLTEXT 1 0 8 "unapply(" }{MPLTEXT 1 0 25 "simplify(dPinfty1dt1inter"
 }{MPLTEXT 1 0 14 "(p1sol,p2sol))" }{MPLTEXT 1 0 7 ",q1,q2)" }{MPLTEXT
 1 0 2 ":\n" }{MPLTEXT 1 0 19 "dPinfty0dt1inter2:=" }{MPLTEXT 1 0 8 "u
napply(" }{MPLTEXT 1 0 25 "simplify(dPinfty0dt1inter" }{MPLTEXT 1 0 
14 "(p1sol,p2sol))" }{MPLTEXT 1 0 7 ",q1,q2)" }{MPLTEXT 1 0 2 ":\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "dQinfty1dt1:=simplify(" }
{MPLTEXT 1 0 32 "dQinfty1dt1inter2(q1sol,q2sol));" }{MPLTEXT 1 0 1 "\n
" }{MPLTEXT 1 0 55 "dQinfty0dt1:=simplify(dQinfty0dt1inter2(q1sol,q2so
l));\n" }{MPLTEXT 1 0 55 "dPinfty1dt1:=simplify(dPinfty1dt1inter2(q1so
l,q2sol));\n" }{MPLTEXT 1 0 54 "dPinfty0dt1:=simplify(dPinfty0dt1inter
2(q1sol,q2sol));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 
1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 91 "dQinfty1dt2inter:=un
apply( simplify( diff(Qinfty1sol,q1)*dq1dt2+diff(Qinfty1sol,q2)*dq2dt2
\n" }{MPLTEXT 1 0 84 "+diff(Qinfty1sol,p1)*dp1dt2+diff(Qinfty1sol,p2)*
dp2dt2+diff(Qinfty1sol,t2)),p1,p2):\n" }{MPLTEXT 1 0 91 "dQinfty0dt2in
ter:=unapply( simplify( diff(Qinfty0sol,q1)*dq1dt2+diff(Qinfty0sol,q2)
*dq2dt2\n" }{MPLTEXT 1 0 84 "+diff(Qinfty0sol,p1)*dp1dt2+diff(Qinfty0s
ol,p2)*dp2dt2+diff(Qinfty0sol,t2)),p1,p2):\n" }{MPLTEXT 1 0 91 "dPinft
y1dt2inter:=unapply( simplify( diff(Pinfty1sol,q1)*dq1dt2+diff(Pinfty1
sol,q2)*dq2dt2\n" }{MPLTEXT 1 0 84 "+diff(Pinfty1sol,p1)*dp1dt2+diff(P
infty1sol,p2)*dp2dt2+diff(Pinfty1sol,t2)),p1,p2):\n" }{MPLTEXT 1 0 91 
"dPinfty0dt2inter:=unapply( simplify( diff(Pinfty0sol,q1)*dq1dt2+diff(
Pinfty0sol,q2)*dq2dt2\n" }{MPLTEXT 1 0 84 "+diff(Pinfty0sol,p1)*dp1dt2
+diff(Pinfty0sol,p2)*dp2dt2+diff(Pinfty0sol,t2)),p1,p2):\n" }{MPLTEXT 
1 0 1 "\n" }{MPLTEXT 1 0 75 "dQinfty1dt2inter2:=unapply(simplify(dQinf
ty1dt2inter(p1sol,p2sol)),q1,q2):\n" }{MPLTEXT 1 0 75 "dQinfty0dt2inte
r2:=unapply(simplify(dQinfty0dt2inter(p1sol,p2sol)),q1,q2):\n" }
{MPLTEXT 1 0 75 "dPinfty1dt2inter2:=unapply(simplify(dPinfty1dt2inter(
p1sol,p2sol)),q1,q2):\n" }{MPLTEXT 1 0 75 "dPinfty0dt2inter2:=unapply(
simplify(dPinfty0dt2inter(p1sol,p2sol)),q1,q2):\n" }{MPLTEXT 1 0 1 "\n
" }{MPLTEXT 1 0 55 "dQinfty1dt2:=simplify(dQinfty1dt2inter2(q1sol,q2so
l));\n" }{MPLTEXT 1 0 55 "dQinfty0dt2:=simplify(dQinfty0dt2inter2(q1so
l,q2sol));\n" }{MPLTEXT 1 0 55 "dPinfty1dt2:=simplify(dPinfty1dt2inter
2(q1sol,q2sol));\n" }{MPLTEXT 1 0 54 "dPinfty0dt2:=simplify(dPinfty0dt
2inter2(q1sol,q2sol));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<$/I(Pinfty0G6\",$*&,&I#p1GF%\"\"\"I#p2GF%!\"\"F*,&I#q1GF
%F*I#q2GF%F,F,F,/I(Pinfty1GF%,$*&,&*&F)F*F.F*F**&F+F*F/F*F,F*F-F,F," }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I&q1solG6\",&*&#\"\"\"\"\"#F(I(Qinfty1
GF$F(!\"\"*&F'F(),&*$)F*F)F(F(*&\"\"%F(I(Qinfty0GF$F(F+F'F(F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I&q2solG6\",&*&#\"\"\"\"\"#F(I(Qinfty1G
F$F(!\"\"*&F'F(),&*$)F*F)F(F(*&\"\"%F(I(Qinfty0GF$F(F+F'F(F(" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I&p1solG6\",&*&I(Pinfty0GF$\"\"\"I#q2GF$F(F(I
(Pinfty1GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&p2solG6\",&*&I(Pi
nfty0GF$\"\"\"I#q1GF$F(F(I(Pinfty1GF$!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I,dQinfty1dt1G6\",$*&\"\"#\"\"\"I(Pinft
y0GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dQinfty0dt1G6\",&*(\"\"#
\"\"\"I(Pinfty0GF$F(I(Qinfty1GF$F(F(*&F'F(I(Pinfty1GF$F(F(" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I,dPinfty1dt1G6\",4*&\"\"&\"\"\")I(Qinfty1GF$\"
\"%F(!\"\"*&,&*&\"\"'F(I#t2GF$F(F,*&\"#7F(I(Qinfty0GF$F(F(F()F*\"\"#F(
F(*(F+F(F*F(I#t1GF$F(F(*$)F1F6F(F,*(F+F(F4F(F1F(F(*$)I(Pinfty0GF$F6F(F
,*&\"\"$F()F4F6F(F,*&F6F(I)tinfty10GF$F(F,F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I,dPinfty0dt1G6\",(*&\"\"%\"\"\")I(Qinfty1GF$\"\"$F(F(*&
,&*&F'F(I#t2GF$F(F(*&\"\"'F(I(Qinfty0GF$F(!\"\"F(F*F(F(*&\"\"#F(I#t1GF
$F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dQinfty1dt2G6\",&*&I(Pinfty0
GF$\"\"\"I(Qinfty1GF$F(F(I(Pinfty1GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I,dQinfty0dt2G6\",**&I(Pinfty0GF$\"\"\")I(Qinfty1GF$\"\"#F(F(*&F'
F(I(Qinfty0GF$F(!\"\"*&I(Pinfty1GF$F(F*F(F(#F(F+F(" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I,dPinfty1dt2G6\",**(\"\"$\"\"\")I(Qinfty0GF$\"\"#F(I(Q
infty1GF$F(F(*&,(*&F+F()F,F'F(!\"\"*(F+F(F,F(I#t2GF$F(F1I#t1GF$F(F(F*F
(F(*&)I(Pinfty0GF$F+F(F,F(F1*&F7F(I(Pinfty1GF$F(F1" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I,dPinfty0dt2G6\",4*&#\"\"\"\"\"#F()I(Qinfty1GF$\"\"%F(
!\"\"*(F'F(,&*&F)F(I#t2GF$F(F-*&\"\"'F(I(Qinfty0GF$F(F(F()F+F)F(F(*&F+
F(I#t1GF$F(F(*&F'F()F1F)F(F-*(F)F(F4F(F1F(F(*&F'F()I(Pinfty0GF$F)F(F(*
&#\"\"$F)F()F4F)F(F-I)tinfty10GF$F-F'F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 218 226 "We compute the associated Hamiltonians. Since the chang
e of coordinates is time-independent and symplectic and since we know \+
that (q,p) are canonical coordinates, we may simply replace the variab
les in the former Hamiltonians." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 69 "Ham1intermediate:=unapply( simplify(Ham1(q1,q2,p1sol,p2sol)),q
1,q2):\n" }{MPLTEXT 1 0 17 "Ham1QP:=simplify(" }{MPLTEXT 1 0 31 "Ham1i
ntermediate(q1sol,q2sol)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 "Ham
1QPbis:=2*(Pinfty0*Pinfty1*omega^2+(1/2)*Pinfty0^2*Qinfty1" }{MPLTEXT 
1 0 191 "*omega+(1/2)*Qinfty1^5/omega^5+(1/2)*(omega*tau2-4*Qinfty0)*Q
infty1^3/omega^4 -(1/2)*Qinfty1^2*tau1/omega^2+3*Qinfty0^2*Qinfty1/2/o
mega^3-Qinfty0*Qinfty1*tau2/omega^2+(1/2)*Qinfty1*(tau2^2/4+" }
{MPLTEXT 1 0 12 "2*tinfty10-1" }{MPLTEXT 1 0 36 ") /omega+(1/2)*Qinfty
0*tau1/omega) ;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "simplify(Ham1Q
P-" }{MPLTEXT 1 0 9 "Ham1QPbis" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 
1 "\n" }{MPLTEXT 1 0 69 "Ham2intermediate:=unapply( simplify(Ham2(q1,q
2,p1sol,p2sol)),q1,q2):\n" }{MPLTEXT 1 0 48 "Ham2QP:=simplify(Ham2inte
rmediate(q1sol,q2sol)):" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 177 "Ham2QPbis:=2*( (1/4)*Pinfty1^2*omega^2+(1/2)*omega*Q
infty1*Pinfty0*Pinfty1-(1/4)*omega*Qinfty0*Pinfty0^2 +(1/4)*Qinfty1^2*
Pinfty0^2+(1/4)*1*Pinfty0*omega+(1/4)*Qinfty0^3/omega^3" }{MPLTEXT 1 
0 130 "-3/4*Qinfty1^2*Qinfty0^2/omega^4-(1/4)*tau2*Qinfty0^2/omega^2+(
1/4)*Qinfty0*Qinfty1^4/omega^5+(1/4)*Qinfty0*tau2*Qinfty1^2/omega^3" }
{MPLTEXT 1 0 53 "-(1/4)*tau1*Qinfty0*Qinfty1/omega^2+Qinfty0*(tau2^2/4
" }{MPLTEXT 1 0 11 "+2*tinfty10" }{MPLTEXT 1 0 16 "-1)/(4*omega));\n" 
}{MPLTEXT 1 0 32 "simplify(series(Ham2QP-Ham2QPbis" }{MPLTEXT 1 0 11 "
,Qinfty1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 
9 "simplify(" }{MPLTEXT 1 0 11 "dQinfty1dt1" }{MPLTEXT 1 0 23 "-diff(H
am1QP,Pinfty1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }
{MPLTEXT 1 0 35 "dQinfty0dt1-diff(Ham1QP,Pinfty0));\n" }{MPLTEXT 1 0 
9 "simplify(" }{MPLTEXT 1 0 38 "dPinfty1dt1-(-diff(Ham1QP,Qinfty1)));
\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 37 "dPinfty0dt1-(-diff(H
am1QP,Qinfty0)));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT
 1 0 9 "simplify(" }{MPLTEXT 1 0 35 "dQinfty1dt2-diff(Ham2QP,Pinfty1))
;\n" }{MPLTEXT 1 0 44 "simplify(dQinfty0dt2-diff(Ham2QP,Pinfty0));\n" 
}{MPLTEXT 1 0 47 "simplify(dPinfty1dt2-(-diff(Ham2QP,Qinfty1)));\n" }
{MPLTEXT 1 0 46 "simplify(dPinfty0dt2-(-diff(Ham2QP,Qinfty0)));" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Ham1QPbisG6\",4
*(\"\"#\"\"\"I(Pinfty0GF$F(I(Pinfty1GF$F(F(*&)F)F'F(I(Qinfty1GF$F(F(*$
)F-\"\"&F(F(*&,&*&F'F(I#t2GF$F(F(*&\"\"%F(I(Qinfty0GF$F(!\"\"F()F-\"\"
$F(F(*(F'F(I#t1GF$F()F-F'F(F8*(F:F()F7F'F(F-F(F(**F6F(F7F(F-F(F4F(F8*&
F-F(,(*$)F4F'F(F(*&F'F(I)tinfty10GF$F(F(F(F8F(F(*(F'F(F7F(F<F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">
I*Ham2QPbisG6\",:*&#\"\"\"\"\"#F()I(Pinfty1GF$F)F(F(*(I(Pinfty0GF$F(F+
F(I(Qinfty1GF$F(F(*(F'F()F-F)F(I(Qinfty0GF$F(!\"\"*(F'F(F0F()F.F)F(F(*
&F'F(F-F(F(*&F'F()F1\"\"$F(F(*(#F8F)F()F1F)F(F4F(F2*&F;F(I#t2GF$F(F2*(
F'F(F1F()F.\"\"%F(F(*(F1F(F4F(F=F(F(*(F1F(F.F(I#t1GF$F(F2*(F'F(F1F(,(*
$)F=F)F(F(*&F)F(I)tinfty10GF$F(F(F(F2F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 218 84 "We may also verify the formula for Hinft
y0 and Hinfty1 appearing in the computation." }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 25 "Hinfty1function:=unapply(" }{MPLTEXT 1 0 15 "Hinf
ty1,p1,p2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 39 "Hinfty1intermediat
e:=unapply( simplify(" }{MPLTEXT 1 0 15 "Hinfty1function" }{MPLTEXT 1 
0 23 "(p1sol,p2sol)),q1,q2):\n" }{MPLTEXT 1 0 7 "Hinfty1" }{MPLTEXT 1 
0 13 "QP:=simplify(" }{MPLTEXT 1 0 7 "Hinfty1" }{MPLTEXT 1 0 27 "inter
mediate(q1sol,q2sol));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 41 "Hinfty0function:=unapply(Hinfty0,p1,p2):\n" }{MPLTEXT
 1 0 77 "Hinfty0intermediate:=unapply( simplify(Hinfty0function(p1sol,
p2sol)),q1,q2):\n" }{MPLTEXT 1 0 54 "Hinfty0QP:=simplify(Hinfty0interm
ediate(q1sol,q2sol));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Hinfty1QPG6
\",.*$)I(Qinfty1GF$\"\"&\"\"\"F**&,&*&\"\"#F*I#t2GF$F*F**&\"\"%F*I(Qin
fty0GF$F*!\"\"F*)F(\"\"$F*F**(F.F*I#t1GF$F*)F(F.F*F3*&,.*$)I(Pinfty0GF
$F.F*F**&F5F*)F2F.F*F**(F1F*F2F*F/F*F3*$)F/F.F*F**&F.F*I)tinfty10GF$F*
F*F*F3F*F(F*F**(F.F*F2F*F7F*F**(F.F*F=F*I(Pinfty1GF$F*F*" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I*Hinfty0QPG6\",0*$)I(Qinfty0GF$\"\"$\"\"\"F**&,&
*&F)F*)I(Qinfty1GF$\"\"#F*!\"\"*&F0F*I#t2GF$F*F1F*)F(F0F*F**&,0*$)F/\"
\"%F*F**(F0F*F.F*F3F*F**$)I(Pinfty0GF$F0F*F1*(F0F*F/F*I#t1GF$F*F1*$)F3
F0F*F**&F0F*I)tinfty10GF$F*F*F*F1F*F(F*F**&F<F*F.F*F***F0F*F=F*I(Pinft
y1GF$F*F/F*F**$)FFF0F*F*F=F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 58 "
We may also rewrite the Lax matrix in the variables (Q,P)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "tdL11function:=unapply( tdL[1,1],p1
,p2):\n" }{MPLTEXT 1 0 28 "tdL11intermediate:=unapply( " }{MPLTEXT 1 
0 32 "tdL11function(p1sol,p2sol),q1,q2" }{MPLTEXT 1 0 3 "):\n" }
{MPLTEXT 1 0 18 "tdL11QP:=simplify(" }{MPLTEXT 1 0 33 "tdL11intermedia
te(q1sol,q2sol));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 41 "tdL12funct
ion:=unapply( tdL[1,2],p1,p2):\n" }{MPLTEXT 1 0 63 "tdL12intermediate:
=unapply( tdL12function(p1sol,p2sol),q1,q2):\n" }{MPLTEXT 1 0 51 "tdL1
2QP:=simplify(tdL12intermediate(q1sol,q2sol));\n" }{MPLTEXT 1 0 1 "\n"
 }{MPLTEXT 1 0 41 "tdL21function:=unapply( tdL[2,1],p1,p2):\n" }
{MPLTEXT 1 0 63 "tdL21intermediate:=unapply( tdL21function(p1sol,p2sol
),q1,q2):\n" }{MPLTEXT 1 0 50 "tdL21QP:=simplify(tdL21intermediate(q1s
ol,q2sol));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(tdL11QPG6\",**$)I'lamb
daGF$\"\"$\"\"\"!\"\"*&,(*$)I(Qinfty1GF$\"\"#F*F*I(Pinfty0GF$F+I(Qinft
y0GF$F+F*F(F*F**&,&F2F+F3F*F*F0F*F*I(Pinfty1GF$F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I(tdL12QPG6\",(*&I(Qinfty1GF$\"\"\"I'lambdaGF$F(F(*$)F)
\"\"#F(F(I(Qinfty0GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(tdL21QPG6
\",8*$)I(Qinfty1GF$\"\"%\"\"\"F**(\"\"#F*)F(\"\"$F*I'lambdaGF$F*!\"\"*
&,**&F,F*)F/F,F*F**&F,F*I(Pinfty0GF$F*F**&F)F*I(Qinfty0GF$F*F0*&F,F*I#
t2GF$F*F*F*)F(F,F*F**&,(*&,&F9F0F7F*F*F/F*F**&F,F*I#t1GF$F*F0*&F,F*I(P
infty1GF$F*F*F*F(F*F**&,(F9F*F5F0*&F,F*F8F*F0F*F4F*F**&,&F@F*FBF0F*F/F
*F**$)F:F,F*F**(F,F*F8F*F:F*F0*$)F6F,F*F0*$)F8F,F*F**&F,F*I)tinfty10GF
$F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 116 "Computation of the spe
ctral invariants and verification of Theorem 5.2 relating Hamiltonians
 and spectral invariants" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"assume(q1>q2):\n" }{MPLTEXT 1 0 91 "lambdaplus:=series(simplify(serie
s(Eigenvalues(tdL)[1],lambda=infinity)),lambda=infinity):\n" }{MPLTEXT
 1 0 81 "CoefflambdaplusLambda3:=simplify(-residue(lambdaplus/lambda^4
,lambda=infinity));\n" }{MPLTEXT 1 0 81 "CoefflambdaplusLambda2:=simpl
ify(-residue(lambdaplus/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 
81 "CoefflambdaplusLambda1:=simplify(-residue(lambdaplus/lambda^2,lamb
da=infinity));\n" }{MPLTEXT 1 0 81 "CoefflambdaplusLambda0:=simplify(-
residue(lambdaplus/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 86 "Co
efflambdaplusLambdaMinus1:=simplify(-residue(lambdaplus/lambda^0,lambd
a=infinity));\n" }{MPLTEXT 1 0 89 "CoefflambdaplusLambdaMinus2:=simpli
fy(-residue(lambdaplus/lambda^(-1),lambda=infinity)):\n" }{MPLTEXT 1 
0 89 "CoefflambdaplusLambdaMinus3:=simplify(-residue(lambdaplus/lambda
^(-2),lambda=infinity)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "I1:
=CoefflambdaplusLambdaMinus2;\n" }{MPLTEXT 1 0 37 "I2:=1/2*Coefflambda
plusLambdaMinus3;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "I1function
:=unapply(I1,p1,p2):\n" }{MPLTEXT 1 0 31 "I2function:=unapply(I2,p1,p2
):\n" }{MPLTEXT 1 0 49 "I1inter:=unapply(I1function(p1sol,p2sol),q1,q2
):\n" }{MPLTEXT 1 0 49 "I2inter:=unapply(I2function(p1sol,p2sol),q1,q2
):\n" }{MPLTEXT 1 0 79 "I1QP:=unapply(simplify(I1inter(q1sol,q2sol)),Q
infty0,Qinfty1,Pinfty0,Pinfty1);\n" }{MPLTEXT 1 0 79 "I2QP:=unapply(si
mplify(I2inter(q1sol,q2sol)),Qinfty0,Qinfty1,Pinfty0,Pinfty1);\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "simplify(Ham1QP-2*I1QP(Qinfty0,Q
infty1,Pinfty0,Pinfty1));\n" }{MPLTEXT 1 0 79 "-residue(lambda^(-2)*td
L[1,2]*diff(tdL[1,1]/tdL[1,2],lambda),lambda=infinity);\n" }{MPLTEXT 
1 0 13 "-Qinfty1sol;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "simplif
y(Ham2QP-2*I2QP(Qinfty0,Qinfty1,Pinfty0,Pinfty1));\n" }{MPLTEXT 1 0 
89 "simplify(-residue(lambda^(-1)*tdL[1,2]*diff(tdL[1,1]/tdL[1,2],lamb
da),lambda=infinity));\n" }{MPLTEXT 1 0 39 "simplify(-1/2*Qinfty0sol+P
infty0sol/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7CoefflambdaplusLambd
a3G6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7CoefflambdaplusLambda
2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7CoefflambdaplusLambda1G
6\"I#t2GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7CoefflambdaplusLambda0G
6\"I#t1GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I<CoefflambdaplusLambdaMi
nus1G6\"I)tinfty10GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#I1G6\"*&,:*$
)I$q1|irG6#/I'_localG%*protectedGI0_n2254191460960GF$\"\"'\"\"\"!\"\"*
(\"\"#F0)F)\"\"%F0I#t2GF$F0F1*(F3F0)F)\"\"$F0I#t1GF$F0F1*&,&*$)F6F3F0F
1*&F3F0I)tinfty10GF$F0F1F0)F)F3F0F0**F3F0F6F0F:F0F)F0F1*$)I$q2|irG6#/F
,I0_n2254191461248GF$F/F0F0*(F3F0)FEF5F0F6F0F0*(F3F0)FEF9F0F:F0F0*&,&F
=F0F?F0F0)FEF3F0F0**F3F0F6F0F:F0FEF0F0*$)I#p1GF$F3F0F0*$)I#p2GF$F3F0F1
F0,&*&F3F0F)F0F0*&F3F0FEF0F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#I2G
6\",$*(#\"\"\"\"\"#F(,.*&)I$q1|irG6#/I'_localG%*protectedGI0_n22541914
60960GF$\"\"'F(I$q2|irG6#/F0I0_n2254191461248GF$F(F(**F)F()F-\"\"%F(F4
F(I#t2GF$F(F(**F)F()F-\"\"$F(F4F(I#t1GF$F(F(*(,&*$)F;F)F(F(*&F)F(I)tin
fty10GF$F(F(F(F4F()F-F)F(F(*&,0*$)F4F3F(!\"\"*(F)F()F4F:F(F;F(FK*(F)F(
)F4F>F(F?F(FK*&,&FBFKFDFKF()F4F)F(F(*$)I#p2GF$F)F(F(*$)F?F)F(FK*(F)F(F
EF(F;F(FKF(F-F(F(*&F4F(,(*$)I#p1GF$F)F(F(FVFK*(F)F(F;F(FEF(FKF(FKF(,&*
&F)F(F-F(F(*&F)F(F4F(FKFKF(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%I1QPG6
\"f*6&I(Qinfty0GF$I(Qinfty1GF$I(Pinfty0GF$I(Pinfty1GF$F$6$I)operatorGF
$I&arrowGF$F$,0*&#\"\"\"\"\"#F1)F(\"\"&F1F1*(F0F1,&*&F2F1I#t2GF$F1F1*&
\"\"%F1F'F1!\"\"F1)F(\"\"$F1F1*&)F(F2F1I#t1GF$F1F;*(F0F1,,*$)F)F2F1F1*
&F=F1)F'F2F1F1*(F:F1F'F1F8F1F;*$)F8F2F1F1*&F2F1I)tinfty10GF$F1F1F1F(F1
F1*&F8F1F@F1F;*&F'F1F@F1F1*&F)F1F*F1F1F$F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%I2QPG6\"f*6&I(Qinfty0GF$I(Qinfty1GF$I(Pinfty0GF$I(Pinf
ty1GF$F$6$I)operatorGF$I&arrowGF$F$,2*&#\"\"\"\"\"%F1)F'\"\"$F1F1*(F0F
1,&*&F4F1)F(\"\"#F1!\"\"*&F9F1I#t2GF$F1F:F1)F'F9F1F1*(F0F1,.*$)F(F2F1F
1*(F9F1F8F1F<F1F1*$)F)F9F1F:*(F9F1F(F1I#t1GF$F1F:*$)F<F9F1F1*&F9F1I)ti
nfty10GF$F1F1F1F'F1F1*(F0F1FDF1F8F1F1**#F1F9F1F)F1F*F1F(F1F1*&F0F1)FFF
9F1F:*(FMF1FJF1F<F1F:*&F0F1)F*F9F1F1F$F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",&*(\"\"#\"\"\"I#t1G6\"F%I#t2GF'F%F%I(Qinfty1GF'!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",&I$q1|irG6#/I'_localG%*protectedGI0_n22
54191460960G6\"\"\"\"I$q2|irG6#/F&I0_n2254191461248GF)F*" }}{PARA 11 "
" 1 "" {XPPMATH 20 ",&I$q1|irG6#/I'_localG%*protectedGI0_n225419146096
0G6\"\"\"\"I$q2|irG6#/F&I0_n2254191461248GF)F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",**&#\"\"\"\"\"#F%I(Qinfty0G6\"F%!\"\"*&F$F%I(Pinfty0GF(F
%F%*&F$F%)I#t1GF(F&F%F%*&I)tinfty10GF(F%I#t2GF(F%F%" }}{PARA 11 "" 1 "
" {XPPMATH 20 "*&,**&)I$q1|irG6#/I'_localG%*protectedGI0_n225419146096
0G6\"\"\"#\"\"\"I$q2|irG6#/F)I0_n2254191461248GF,F.!\"\"*&F&F.)F/F-F.F
.I#p1GF,F3I#p2GF,F.F.,&F&F.F/F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,
**&)I$q1|irG6#/I'_localG%*protectedGI0_n2254191460960G6\"\"\"#\"\"\"I$
q2|irG6#/F)I0_n2254191461248GF,F.!\"\"*&F&F.)F/F-F.F.I#p1GF,F3I#p2GF,F
.F.,&*&F-F.F&F.F.*&F-F.F/F.F3F3" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 
218 59 "Second change of Darboux coordinates: From (Q,P) to (Q,R). " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 145 "Definition of the change of co
ordinates and its inverse. This change is time-independent bu not symp
lectic so the evolutions are not canonically " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "Pinfty0sol2:=Qinfty1^2-Qinfty0-Rinfty1;\n" }
{MPLTEXT 1 0 68 "Pinfty1sol2 :=-Qinfty1^3+2*Qinfty0*Qinfty1+Qinfty1*Ri
nfty1-Rinfty0;\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 18 "Rinfty
1-( -Pinfty0" }{MPLTEXT 1 0 4 "sol2" }{MPLTEXT 1 0 22 "-Qinfty0+Qinfty
1^2));\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 58 "Rinfty0-(-Pinf
ty1sol2-Pinfty0sol2*Qinfty1+Qinfty0*Qinfty1)" }{MPLTEXT 1 0 3 ");\n" }
{MPLTEXT 1 0 12 "Rinfty1sol:=" }{MPLTEXT 1 0 8 "-Pinfty0" }{MPLTEXT 1 
0 20 "-Qinfty0+Qinfty1^2;\n" }{MPLTEXT 1 0 12 "Rinfty0sol:=" }{MPLTEXT
 1 0 41 "-Pinfty1-Pinfty0*Qinfty1+Qinfty0*Qinfty1;" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I,Pinfty0sol2G6\",(*$)I(Qinfty1GF$\"\"#\"\"\"F*I(Qinfty
0GF$!\"\"I(Rinfty1GF$F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,Pinfty1sol
2G6\",**$)I(Qinfty1GF$\"\"$\"\"\"!\"\"*(\"\"#F*I(Qinfty0GF$F*F(F*F**&F
(F*I(Rinfty1GF$F*F*I(Rinfty0GF$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I+Rinfty1solG6\",(*$)I(Qinfty1GF$\"\"#\"\"\"F*I(Pinfty0G
F$!\"\"I(Qinfty0GF$F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Rinfty0solG6
\",(*&I(Pinfty0GF$\"\"\"I(Qinfty1GF$F(!\"\"*&I(Qinfty0GF$F(F)F(F(I(Pin
fty1GF$F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 54 "Expression of the L
ax matrix in the coordinates (Q,R)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "tdL11QPfunction:=unapply( " }{MPLTEXT 1 0 7 "tdL11QP"
 }{MPLTEXT 1 0 19 ",Pinfty0,Pinfty1):\n" }{MPLTEXT 1 0 35 "tdL11QR:=si
mplify( tdL11QPfunction(" }{MPLTEXT 1 0 11 "Pinfty0sol2" }{MPLTEXT 1 
0 1 "," }{MPLTEXT 1 0 11 "Pinfty1sol2" }{MPLTEXT 1 0 4 "));\n" }
{MPLTEXT 1 0 52 "tdL12QPfunction:=unapply( tdL12QP,Pinfty0,Pinfty1):\n
" }{MPLTEXT 1 0 58 "tdL12QR:=simplify( tdL12QPfunction(Pinfty0sol2,Pin
fty1sol2" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 52 "tdL21QPfunction:=u
napply( tdL21QP,Pinfty0,Pinfty1):\n" }{MPLTEXT 1 0 58 "tdL21QR:=simpli
fy( tdL21QPfunction(Pinfty0sol2,Pinfty1sol2" }{MPLTEXT 1 0 3 "));" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I(tdL11QRG6\",(*$)I'lambdaGF$\"\"$\"\"
\"!\"\"*&I(Rinfty1GF$F*F(F*F*I(Rinfty0GF$F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I(tdL12QRG6\",(*&I(Qinfty1GF$\"\"\"I'lambdaGF$F(F(*$)F)
\"\"#F(F(I(Qinfty0GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(tdL21QRG6
\",4*&,&*&\"\"#\"\"\"I#t2GF$F*F**&F)F*I(Rinfty1GF$F*F*F*)I(Qinfty1GF$F
)F*F**&,(*&,&F(!\"\"F,F4F*I'lambdaGF$F*F**&F)F*I#t1GF$F*F4*&F)F*I(Rinf
ty0GF$F*F4F*F/F*F**&F'F*)F5F)F*F**&,&F6F*F8F*F*F5F*F**$)F+F)F*F**(F)F*
I(Qinfty0GF$F*F+F*F4*(F)F*FAF*F-F*F4*$)F-F)F*F4*&F)F*I)tinfty10GF$F*F*
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 54 "Computation of the evolution
s of Rinfty0 and Rinfty1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "dRinfty1dt1inter:=unapply( simplify( diff(" }{MPLTEXT 1 0 20 "Rinf
ty1sol,Qinfty1)*" }{MPLTEXT 1 0 12 "dQinfty1dt1+" }{MPLTEXT 1 0 1 " " 
}{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 20 "Rinfty1sol,Qinfty0)*" }
{MPLTEXT 1 0 12 "dQinfty0dt1+" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 
20 "Rinfty1sol,Pinfty1)*" }{MPLTEXT 1 0 12 "dPinfty1dt1+" }{MPLTEXT 1 
0 5 "diff(" }{MPLTEXT 1 0 20 "Rinfty1sol,Pinfty0)*" }{MPLTEXT 1 0 12 "
dPinfty0dt1+" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 35 "Rinfty1sol,t1)
 ),Pinfty0,Pinfty1):\n" }{MPLTEXT 1 0 11 "dRinfty0dt1" }{MPLTEXT 1 0 
5 "inter" }{MPLTEXT 1 0 2 ":=" }{MPLTEXT 1 0 8 "unapply(" }{MPLTEXT 1 
0 15 "simplify( diff(" }{MPLTEXT 1 0 149 "Rinfty0sol,Qinfty1)*dQinfty1
dt1+ diff(Rinfty0sol,Qinfty0)*dQinfty0dt1+diff(Rinfty0sol,Pinfty1)*dPi
nfty1dt1+diff(Rinfty0sol,Pinfty0)*dPinfty0dt1+diff(" }{MPLTEXT 1 0 35 
"Rinfty0sol,t1) ),Pinfty0,Pinfty1):\n" }{MPLTEXT 1 0 26 "dQinfty1dt1in
ter:=unapply(" }{MPLTEXT 1 0 11 "dQinfty1dt1" }{MPLTEXT 1 0 19 " ,Pinf
ty0,Pinfty1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "dQinfty0dt1inter
:=unapply(dQinfty0dt1 ,Pinfty0,Pinfty1):" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "dRinfty1dt1:=simplify(" }
{MPLTEXT 1 0 17 "dRinfty1dt1inter(" }{MPLTEXT 1 0 12 "Pinfty0sol2," }
{MPLTEXT 1 0 14 "Pinfty1sol2));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 
66 "dRinfty0dt1:=simplify(dRinfty0dt1inter(Pinfty0sol2,Pinfty1sol2));
\n" }{MPLTEXT 1 0 68 "dQinfty1QRdt1:=simplify(dQinfty1dt1inter(Pinfty0
sol2,Pinfty1sol2));\n" }{MPLTEXT 1 0 67 "dQinfty0QRdt1:=simplify(dQinf
ty0dt1inter(Pinfty0sol2,Pinfty1sol2));" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I,dRinfty1dt1G6\",(*&,&*&\"\"%\"\"\"I#t2GF$F*!\"\"*&F)F*I(Rinfty1GF
$F*F,F*I(Qinfty1GF$F*F**&\"\"#F*I#t1GF$F*F**&F1F*I(Rinfty0GF$F*F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I,dRinfty0dt1G6\",2*&,&*&\"\"#\"\"\"I#t
2GF$F*F**&F)F*I(Rinfty1GF$F*F*F*)I(Qinfty1GF$F)F*F**&,&*&F)F*I#t1GF$F*
!\"\"*&F)F*I(Rinfty0GF$F*F4F*F/F*F**$)F+F)F*F**(\"\"%F*I(Qinfty0GF$F*F
+F*F4*(F:F*F;F*F-F*F4*$)F-F)F*F4*&F)F*I)tinfty10GF$F*F*F*F4" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I.dQinfty1QRdt1G6\",(*&\"\"#\"\"\")I(Qinfty1G
F$F'F(F(*&F'F(I(Qinfty0GF$F(!\"\"*&F'F(I(Rinfty1GF$F(F-" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I.dQinfty0QRdt1G6\",&*(\"\"#\"\"\"I(Qinfty0GF$F(I(
Qinfty1GF$F(F(*&F'F(I(Rinfty0GF$F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 218 61 "Rewriting of the quantities in terms of the coordinates \+
(Q,R)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "tdA1function11:=un
apply(tdA1[1,1],p1,p2):\n" }{MPLTEXT 1 0 42 "tdA1function12:=unapply(t
dA1[1,2],p1,p2):\n" }{MPLTEXT 1 0 42 "tdA1function21:=unapply(tdA1[2,1
],p1,p2):\n" }{MPLTEXT 1 0 42 "tdA1function22:=unapply(tdA1[2,2],p1,p2
):\n" }{MPLTEXT 1 0 66 "tdA1QPintermediate11:=unapply(tdA1function11(p
1sol,p2sol),q1,q2):\n" }{MPLTEXT 1 0 66 "tdA1QPintermediate12:=unapply
(tdA1function12(p1sol,p2sol),q1,q2):\n" }{MPLTEXT 1 0 66 "tdA1QPinterm
ediate21:=unapply(tdA1function21(p1sol,p2sol),q1,q2):\n" }{MPLTEXT 1 
0 66 "tdA1QPintermediate22:=unapply(tdA1function22(p1sol,p2sol),q1,q2)
:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "tdA1QP:=Matrix(2,2,0):\n" 
}{MPLTEXT 1 0 58 "tdA1QP[1,1]:=simplify(tdA1QPintermediate11(q1sol,q2s
ol)):\n" }{MPLTEXT 1 0 58 "tdA1QP[1,2]:=simplify(tdA1QPintermediate12(
q1sol,q2sol)):\n" }{MPLTEXT 1 0 58 "tdA1QP[2,1]:=simplify(tdA1QPinterm
ediate21(q1sol,q2sol)):\n" }{MPLTEXT 1 0 58 "tdA1QP[2,2]:=simplify(tdA
1QPintermediate22(q1sol,q2sol)):\n" }{MPLTEXT 1 0 8 "tdA1QP;\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "tdA2function11:=unapply(tdA2[1,1
],p1,p2):\n" }{MPLTEXT 1 0 42 "tdA2function12:=unapply(tdA2[1,2],p1,p2
):\n" }{MPLTEXT 1 0 42 "tdA2function21:=unapply(tdA2[2,1],p1,p2):\n" }
{MPLTEXT 1 0 42 "tdA2function22:=unapply(tdA2[2,2],p1,p2):\n" }
{MPLTEXT 1 0 66 "tdA2QPintermediate11:=unapply(tdA2function11(p1sol,p2
sol),q1,q2):\n" }{MPLTEXT 1 0 66 "tdA2QPintermediate12:=unapply(tdA2fu
nction12(p1sol,p2sol),q1,q2):\n" }{MPLTEXT 1 0 66 "tdA2QPintermediate2
1:=unapply(tdA2function21(p1sol,p2sol),q1,q2):\n" }{MPLTEXT 1 0 66 "td
A2QPintermediate22:=unapply(tdA2function22(p1sol,p2sol),q1,q2):\n" }
{MPLTEXT 1 0 23 "tdA2QP:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 58 "tdA2QP[1,
1]:=simplify(tdA2QPintermediate11(q1sol,q2sol)):\n" }{MPLTEXT 1 0 58 "
tdA2QP[1,2]:=simplify(tdA2QPintermediate12(q1sol,q2sol)):\n" }{MPLTEXT
 1 0 58 "tdA2QP[2,1]:=simplify(tdA2QPintermediate21(q1sol,q2sol)):\n" 
}{MPLTEXT 1 0 58 "tdA2QP[2,2]:=simplify(tdA2QPintermediate22(q1sol,q2s
ol)):\n" }{MPLTEXT 1 0 8 "tdA2QP;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 
0 56 "tdA1QRfunction11:=unapply(tdA1QP[1,1],Pinfty0,Pinfty1):\n" }
{MPLTEXT 1 0 56 "tdA1QRfunction12:=unapply(tdA1QP[1,2],Pinfty0,Pinfty1
):\n" }{MPLTEXT 1 0 56 "tdA1QRfunction21:=unapply(tdA1QP[2,1],Pinfty0,
Pinfty1):\n" }{MPLTEXT 1 0 56 "tdA1QRfunction22:=unapply(tdA1QP[2,2],P
infty0,Pinfty1):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "tdA1QR:=Mat
rix(2,2,0):\n" }{MPLTEXT 1 0 66 "tdA1QR[1,1]:=simplify(tdA1QRfunction1
1(Pinfty0sol2,Pinfty1sol2)):\n" }{MPLTEXT 1 0 66 "tdA1QR[1,2]:=simplif
y(tdA1QRfunction12(Pinfty0sol2,Pinfty1sol2)):\n" }{MPLTEXT 1 0 66 "tdA
1QR[2,1]:=simplify(tdA1QRfunction21(Pinfty0sol2,Pinfty1sol2)):\n" }
{MPLTEXT 1 0 66 "tdA1QR[2,2]:=simplify(tdA1QRfunction22(Pinfty0sol2,Pi
nfty1sol2)):\n" }{MPLTEXT 1 0 8 "tdA1QR;\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 56 "tdA2QRfunction11:=unapply(tdA2QP[1,1],Pinfty0,Pinfty1
):\n" }{MPLTEXT 1 0 56 "tdA2QRfunction12:=unapply(tdA2QP[1,2],Pinfty0,
Pinfty1):\n" }{MPLTEXT 1 0 56 "tdA2QRfunction21:=unapply(tdA2QP[2,1],P
infty0,Pinfty1):\n" }{MPLTEXT 1 0 56 "tdA2QRfunction22:=unapply(tdA2QP
[2,2],Pinfty0,Pinfty1):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "tdA2
QR:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 66 "tdA2QR[1,1]:=simplify(tdA2QRfu
nction11(Pinfty0sol2,Pinfty1sol2)):\n" }{MPLTEXT 1 0 66 "tdA2QR[1,2]:=
simplify(tdA2QRfunction12(Pinfty0sol2,Pinfty1sol2)):\n" }{MPLTEXT 1 0 
66 "tdA2QR[2,1]:=simplify(tdA2QRfunction21(Pinfty0sol2,Pinfty1sol2)):
\n" }{MPLTEXT 1 0 66 "tdA2QR[2,2]:=simplify(tdA2QRfunction22(Pinfty0so
l2,Pinfty1sol2)):\n" }{MPLTEXT 1 0 8 "tdA2QR;\n" }{MPLTEXT 1 0 1 "\n" 
}{MPLTEXT 1 0 22 "tdLQR:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 21 "tdLQR[1,1
]:=tdL11QR:\n" }{MPLTEXT 1 0 21 "tdLQR[1,2]:=tdL12QR:\n" }{MPLTEXT 1 
0 31 "tdLQR[2,1]:=simplify(tdL21QR):\n" }{MPLTEXT 1 0 25 "tdLQR[2,2]:=
-tdLQR[1,1]:\n" }{MPLTEXT 1 0 6 "tdLQR;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowG
F$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6%-I#miGF$6&Q(Qinfty1F'/%'i
talicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moG
F$6.Q(&minus;F'F@/FDQ'normalF'/%&fenceGQ&falseF'/%*separatorGFN/%)stre
tchyGFN/%*symmetricGFN/%(largeopGFN/%.movablelimitsGFN/%'accentGFN/%'l
spaceGQ,0.2222222emF'/%'rspaceGFgn-F:6&Q)&lambda;F'/F>FNF@FJ/%)rowalig
nGQ!F'/%,columnalignGF`o/%+groupalignGF`o/%(rowspanGQ\"1F'/%+columnspa
nGFgo-F56(-I#mnGF$6%FgoF@FJF^oFaoFcoFeoFhoF^oFaoFco-F26'-F56(-F,6)-F,6
%-F]p6%Q\"2F'F@FJ-FG6-Q1&InvisibleTimes;F'FJFLFOFQFSFUFWFY/FfnQ&0.0emF
'/FinF^q-I%msupGF$6%F9Fgp/%1superscriptshiftGQ\"0F'FF-F,6%FgpFjp-F:6&Q
(Pinfty0F'F=F@FCFF-F,6%FgpFjp-F:6&Q(Qinfty0F'F=F@FC-FG6.Q\"+F'F@FJFLFO
FQFSFUFWFYFenFhn-F,6%FgpFjp-F:6&Q#t2F'F=F@FCF^oFaoFcoFeoFho-F56(-F,6&-
FG6.Q*&uminus0;F'F@FJFLFOFQFSFUFWFYFenFhnF9F`rFjnF^oFaoFcoFeoFhoF^oFao
Fco/%&alignGQ%axisF'/F_oQ)baselineF'/FboQ'centerF'/FdoQ'|frleft|hrF'/%
/alignmentscopeGF?/%,columnwidthGQ%autoF'/%&widthGF\\t/%+rowspacingGQ&
1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFg
t/%&frameGFgt/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFN/%-equalcol
umnsGFN/%-displaystyleGFN/%%sideGQ&rightF'/%0minlabelspacingGFdtF@FJ/%
%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfence
dG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtable
GF$66-I$mtrGF$6'-I$mtdGF$6(-F,6'-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'/%+f
oregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'-F?6%Q\"2F'FBFE/%.lineth
icknessGFA/%+denomalignGQ'centerF'/%)numalignGFO/%)bevelledGQ&falseF'F
B-I#moGF$6-Q1&InvisibleTimes;F'FE/%&fenceGFT/%*separatorGFT/%)stretchy
GFT/%*symmetricGFT/%(largeopGFT/%.movablelimitsGFT/%'accentGFT/%'lspac
eGQ&0.0emF'/%'rspaceGFco-F,6#-I%msupGF$6%-I#miGF$6&Q(Qinfty1F'/%'itali
cGQ%trueF'FB/FFQ'italicF'FH/%1superscriptshiftGQ\"0F'-FV6.Q(&minus;F'F
BFEFYFenFgnFinF[oF]oF_o/FboQ,0.2222222emF'/FeoF[q-F,6%F;FU-F,6#-Fio6%-
F\\p6&Q)&lambda;F'/F`pFTFBFEFHFdpFgp-F,6%F;FU-F,6#-F\\p6&Q(Pinfty0F'F_
pFBFbp/%)rowalignGQ!F'/%,columnalignGF`r/%+groupalignGF`r/%(rowspanGFA
/%+columnspanGFA-F56(-F,6%-F,6%F;FU-F,6#Fcq-FV6.Q\"+F'FBFEFYFenFgnFinF
[oF]oF_oFjpF\\q-F,6%F;FU-F,6#F[pF^rFarFcrFerFgrF^rFarFcr-F26'-F56(-F,6
.-FV6.Q*&uminus0;F'FBFEFYFenFgnFinF[oF]oF_oFjpF\\q-F,6#-Fio6%F[p-F?6%Q
\"3F'FBFEFdpFas-F,6%FhoFUFcqFas-F,6%-F#6%-F,6&F^t-F\\p6&Q#t2F'F_pFBFbp
Fas-F,6%FHFU-F\\p6&Q(Qinfty0F'F_pFBFbpFBFEFUF[pFas-F,6%-F#6%-F,6'F`uFg
pF[rFgpFeuFBFEFUFcqFas-F\\p6&Q#t1F'F_pFBFbpFgp-F\\p6&Q(Pinfty1F'F_pFBF
bpF^rFarFcrFerFgr-F56(-F,6(F^tF9FasF]qFasFgqF^rFarFcrFerFgrF^rFarFcr/%
&alignGQ%axisF'/F_rQ)baselineF'/FbrFO/FdrQ'|frleft|hrF'/%/alignmentsco
peGFap/%,columnwidthGQ%autoF'/%&widthGFdw/%+rowspacingGQ&1.0exF'/%.col
umnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF_x/%&frameGF_x
/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFT/%-equalcolumnsGFT/%-dis
playstyleGFT/%%sideGQ&rightF'/%0minlabelspacingGF\\xFBFE/%%openGQ\"[F'
/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modul
enameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrG
F$6'-I$mtdGF$6(-F,6%-I#miGF$6&Q(Qinfty1F'/%'italicGQ%trueF'/%+foregrou
ndGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6.Q(&minus;F'F@/FDQ'nor
malF'/%&fenceGQ&falseF'/%*separatorGFN/%)stretchyGFN/%*symmetricGFN/%(
largeopGFN/%.movablelimitsGFN/%'accentGFN/%'lspaceGQ,0.2222222emF'/%'r
spaceGFgn-F:6&Q)&lambda;F'/F>FNF@FJ/%)rowalignGQ!F'/%,columnalignGF`o/
%+groupalignGF`o/%(rowspanGQ\"1F'/%+columnspanGFgo-F56(-I#mnGF$6%FgoF@
FJF^oFaoFcoFeoFhoF^oFaoFco-F26'-F56(-F,6%-F,6%-F]p6%Q\"2F'F@FJ-FG6-Q1&
InvisibleTimes;F'FJFLFOFQFSFUFWFY/FfnQ&0.0emF'/FinF^q-F:6&Q#t2F'F=F@FC
-FG6.Q\"+F'F@FJFLFOFQFSFUFWFYFenFhn-F,6%FgpFjp-F:6&Q(Rinfty1F'F=F@FCF^
oFaoFcoFeoFho-F56(-F,6&-FG6.Q*&uminus0;F'F@FJFLFOFQFSFUFWFYFenFhnF9Fcq
FjnF^oFaoFcoFeoFhoF^oFaoFco/%&alignGQ%axisF'/F_oQ)baselineF'/FboQ'cent
erF'/FdoQ'|frleft|hrF'/%/alignmentscopeGF?/%,columnwidthGQ%autoF'/%&wi
dthGF_s/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%
noneF'/%,columnlinesGFjs/%&frameGFjs/%-framespacingGQ,0.4em~0.5exF'/%*
equalrowsGFN/%-equalcolumnsGFN/%-displaystyleGFN/%%sideGQ&rightF'/%0mi
nlabelspacingGFgsF@FJ/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'
6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6(-I#moGF$6.Q*&um
inus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&f
alseF'/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/%.mova
blelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222emF'/%'rspaceGFT-F,6%-I&m
fracGF$6)-I#mnGF$6%Q\"1F'F=F@-Fgn6%Q\"2F'F=F@/%.linethicknessGFin/%+de
nomalignGQ'centerF'/%)numalignGFao/%)bevelledGFEF=-F:6-Q1&InvisibleTim
es;F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFjo-F,6#-I%msupGF$6%-I#miGF$6&Q)&l
ambda;F'/%'italicGFEF=F@Fjn/%1superscriptshiftGQ\"0F'-F:6.Q\"+F'F=F@FC
FFFHFJFLFNFPFRFU-F,6%FYFfo-F,6#-Fbp6&Q(Qinfty0F'/FfpQ%trueF'F=/FAQ'ita
licF'Fjp-F,6%FYFfo-F,6#-Fbp6&Q(Rinfty1F'FdqF=Ffq/%)rowalignGQ!F'/%,col
umnalignGFar/%+groupalignGFar/%(rowspanGFin/%+columnspanGFin-F56(-F,6%
-F,6%FYFfo-F,6#FapFjp-F,6%FYFfo-F,6#-Fbp6&Q(Qinfty1F'FdqF=FfqF_rFbrFdr
FfrFhrF_rFbrFdr-F26'-F56(-F,6+-F,6%-F#6%-F,6&F9-Fbp6&Q#t2F'FdqF=Ffq-F:
6.Q(&minus;F'F=F@FCFFFHFJFLFNFPFRFUF\\rF=F@FfoFfsFjp-F,6%FapFfoFetFjp-
F,6%F\\rFfoFapFjp-Fbp6&Q#t1F'FdqF=FfqFjp-Fbp6&Q(Rinfty0F'FdqF=FfqF_rFb
rFdrFfrFhr-F56(-F,6'FWFhtF]qFhtFhqF_rFbrFdrFfrFhrF_rFbrFdr/%&alignGQ%a
xisF'/F`rQ)baselineF'/FcrFao/FerQ'|frleft|hrF'/%/alignmentscopeGFeq/%,
columnwidthGQ%autoF'/%&widthGFev/%+rowspacingGQ&1.0exF'/%.columnspacin
gGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF`w/%&frameGF`w/%-frames
pacingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-equalcolumnsGFE/%-displaystyle
GFE/%%sideGQ&rightF'/%0minlabelspacingGF]wF=F@/%%openGQ\"[F'/%&closeGQ
\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I
,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtd
GF$6(-F,6(-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvaria
ntGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*symmetr
icGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222
emF'/%'rspaceGFT-F,6#-I%msupGF$6%-I#miGF$6&Q)&lambda;F'/%'italicGFEF=F
@-I#mnGF$6%Q\"3F'F=F@/%1superscriptshiftGQ\"0F'-F:6.Q\"+F'F=F@FCFFFHFJ
FLFNFPFRFU-F,6%-Fgn6&Q(Rinfty1F'/F[oQ%trueF'F=/FAQ'italicF'-F:6-Q1&Inv
isibleTimes;F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFcpFfnFco-Fgn6&Q(Rinfty0F
'F[pF=F]p/%)rowalignGQ!F'/%,columnalignGFjp/%+groupalignGFjp/%(rowspan
GQ\"1F'/%+columnspanGFaq-F56(-F,6'-F,6%-Fgn6&Q(Qinfty1F'F[pF=F]pF_pFfn
Fco-F,6#-FZ6%Ffn-F]o6%Q\"2F'F=F@F`oFco-Fgn6&Q(Qinfty0F'F[pF=F]pFhpF[qF
]qF_qFbqFhpF[qF]q-F26'-F56(-F,63-F,6%-F#6%-F,6%-F,6%FarF_p-Fgn6&Q#t2F'
F[pF=F]pFco-F,6%FarF_pFhoF=F@F_p-FZ6%FjqFarF`oFco-F,6%-F#6%-F,6'-F,6%-
F#6%-F,6&F9Fcs-F:6.Q(&minus;F'F=F@FCFFFHFJFLFNFPFRFUFhsF=F@F_pFfnFht-F
,6%FarF_p-Fgn6&Q#t1F'F[pF=F]pFht-F,6%FarF_pFepF=F@F_pFjqFco-F,6%F_sF_p
F_rFco-F,6%-F#6%-F,6%F[uFcoF`uF=F@F_pFfnFco-F,6#-FZ6%FesFarF`oFht-F,6'
FarF_pFdrF_pFesFht-F,6'FarF_pFdrF_pFhoFht-F,6#-FZ6%FhoFarF`oFco-F,6%Fa
rF_p-Fgn6&Q)tinfty10F'F[pF=F]pFhpF[qF]qF_qFbq-F56(-F,6'FWFhtFfoFhtFepF
hpF[qF]qF_qFbqFhpF[qF]q/%&alignGQ%axisF'/FipQ)baselineF'/F\\qQ'centerF
'/F^qQ'|frleft|hrF'/%/alignmentscopeGF\\p/%,columnwidthGQ%autoF'/%&wid
thGF\\x/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%
noneF'/%,columnlinesGFgx/%&frameGFgx/%-framespacingGQ,0.4em~0.5exF'/%*
equalrowsGFE/%-equalcolumnsGFE/%-displaystyleGFE/%%sideGQ&rightF'/%0mi
nlabelspacingGFdxF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 95 "lambdaplusQR:=series(simplify(series(Eigenv
alues(tdLQR)[1],lambda=infinity)),lambda=infinity):\n" }{MPLTEXT 1 0 
85 "CoefflambdaplusLambda3QR:=simplify(-residue(lambdaplusQR/lambda^4,
lambda=infinity));\n" }{MPLTEXT 1 0 85 "CoefflambdaplusLambda2QR:=simp
lify(-residue(lambdaplusQR/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 
0 85 "CoefflambdaplusLambda1QR:=simplify(-residue(lambdaplusQR/lambda^
2,lambda=infinity));\n" }{MPLTEXT 1 0 85 "CoefflambdaplusLambda0QR:=si
mplify(-residue(lambdaplusQR/lambda^1,lambda=infinity));\n" }{MPLTEXT 
1 0 90 "CoefflambdaplusLambdaMinus1QR:=simplify(-residue(lambdaplusQR/
lambda^0,lambda=infinity));\n" }{MPLTEXT 1 0 93 "CoefflambdaplusLambda
Minus2QR:=simplify(-residue(lambdaplusQR/lambda^(-1),lambda=infinity))
:\n" }{MPLTEXT 1 0 93 "CoefflambdaplusLambdaMinus3QR:=simplify(-residu
e(lambdaplusQR/lambda^(-2),lambda=infinity)):\n" }{MPLTEXT 1 0 1 "\n" 
}{MPLTEXT 1 0 45 "I1QR:=expand(CoefflambdaplusLambdaMinus2QR);\n" }
{MPLTEXT 1 0 48 "I2QR:=expand(1/2*CoefflambdaplusLambdaMinus3QR);" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I9CoefflambdaplusLambda3QRG6\"\"\"\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I9CoefflambdaplusLambda2QRG6\"\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I9CoefflambdaplusLambda1QRG6\"I#t2GF$"
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9CoefflambdaplusLambda0QRG6\"I#t1GF
$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I>CoefflambdaplusLambdaMinus1QRG6
\"I)tinfty10GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%I1QRG6\",<*&)I(Qin
fty1GF$\"\"$\"\"\"I(Rinfty1GF$F*F**&I#t2GF$F*F'F*F***\"\"#F*I(Qinfty0G
F$F*F(F*F+F*!\"\"**F/F*F0F*F(F*F-F*F1*&)F(F/F*I(Rinfty0GF$F*F1*&I#t1GF
$F*F4F*F1*(#F*F/F*F(F*)F+F/F*F1*(F9F*)F-F/F*F(F*F**&F0F*F5F*F**&F0F*F7
F*F**&I)tinfty10GF$F*F(F*F**&F5F*F+F*F**&F-F*F7F*F1" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I%I2QRG6\",:*(#\"\"\"\"\"#F()I(Qinfty0GF$F)F(I(Rinfty1
GF$F(!\"\"*(F'F(F*F(I#t2GF$F(F-**F'F(F+F()I(Qinfty1GF$F)F(F,F(F(**F'F(
F+F(F1F(F/F(F(**F'F(F+F(F2F(I(Rinfty0GF$F(F-**F'F(F+F(F2F(I#t1GF$F(F-*
(#F(\"\"%F(F+F()F,F)F(F-*(F9F(F+F()F/F)F(F(*(F'F(F+F(I)tinfty10GF$F(F(
*&F9F()F7F)F(F-*(F'F(F?F(F/F(F-*&F9F()F5F)F(F(" }}}{EXCHG {PARA 211 ""
 0 "" {TEXT 218 56 "Third change of Darboux coordinates: from (Q,R) to
 (u,v)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 41 "Definition of the isos
pectral coordinates" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Qinf
ty1:=uinfty1;\n" }{MPLTEXT 1 0 23 "Qinfty0:=uinfty0+t2/2;\n" }{MPLTEXT
 1 0 22 "Rinfty1:=-t2+vinfty1;\n" }{MPLTEXT 1 0 21 "Rinfty0:=-t1+vinft
y0;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "tdLuv:=simplify(" }
{MPLTEXT 1 0 8 "tdLQR);\n" }{MPLTEXT 1 0 22 "I1uv:=simplify(I1QR);\n" 
}{MPLTEXT 1 0 21 "I2uv:=simplify(I2QR);" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I(Qinfty1G6\"I(uinfty1GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(Qin
fty0G6\",&I(uinfty0GF$\"\"\"*&#F'\"\"#F'I#t2GF$F'F'" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I(Rinfty1G6\",&I#t2GF$!\"\"I(vinfty1GF$\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I(Rinfty0G6\",&I#t1GF$!\"\"I(vinfty0GF$
\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"
I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mt
dGF$6(-F,6*-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvari
antGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*symmet
ricGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.222222
2emF'/%'rspaceGFT-F,6#-I%msupGF$6%-I#miGF$6&Q)&lambda;F'/%'italicGFEF=
F@-I#mnGF$6%Q\"3F'F=F@/%1superscriptshiftGQ\"0F'-F:6.Q\"+F'F=F@FCFFFHF
JFLFNFPFRFU-F,6%-F#6%-F,6&F9-Fgn6&Q#t2F'/F[oQ%trueF'F=/FAQ'italicF'Fco
-Fgn6&Q(vinfty1F'F_pF=FapF=F@-F:6-Q1&InvisibleTimes;F'F@FCFFFHFJFLFNFP
/FSQ&0.0emF'/FVFjpFfn-F:6.Q(&minus;F'F=F@FCFFFHFJFLFNFPFRFU-Fgn6&Q#t1F
'F_pF=FapFco-Fgn6&Q(vinfty0F'F_pF=Fap/%)rowalignGQ!F'/%,columnalignGFg
q/%+groupalignGFgq/%(rowspanGQ\"1F'/%+columnspanGF^r-F56(-F,6)-F,6%-Fg
n6&Q(uinfty1F'F_pF=FapFfpFfnFco-F,6#-FZ6%Ffn-F]o6%Q\"2F'F=F@F`oFco-Fgn
6&Q(uinfty0F'F_pF=FapFco-F,6%-I&mfracGF$6)-F]o6%F^rF=F@F^s/%.linethick
nessGF^r/%+denomalignGQ'centerF'/%)numalignGF_t/%)bevelledGFEF=Ffp-F,6
#F\\pFeqFhqFjqF\\rF_rFeqFhqFjq-F26'-F56(-F,6,F9-F,6#-FZ6%FcpF^sF`oFco-
F,6%-F#6%-F,6+-F,6%F^sFfpF\\sF\\q-F,6'F^sFfpFfnFfpFgrFco-F,6%F^sFfp-FZ
6%FgrF^sF`oFcoF\\pF\\q-F,6%F^sFfpFasF=F@FfpFcpFco-F,6'F^sFfpFbqFfpFfnF
\\q-F,6'F^sFfpFgrFfpFbqFco-F,6%F^sFfp-Fgn6&Q)tinfty10F'F_pF=FapFeqFhqF
jqF\\rF_r-F56(-F,6)FWFco-F,6%-F#6%-F,6%F\\pF\\qFcpF=F@FfpFfnFcoF_qF\\q
FbqFeqFhqFjqF\\rF_rFeqFhqFjq/%&alignGQ%axisF'/FfqQ)baselineF'/FiqF_t/F
[rQ'|frleft|hrF'/%/alignmentscopeGF`p/%,columnwidthGQ%autoF'/%&widthGF
_x/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF
'/%,columnlinesGFjx/%&frameGFjx/%-framespacingGQ,0.4em~0.5exF'/%*equal
rowsGFE/%-equalcolumnsGFE/%-displaystyleGFE/%%sideGQ&rightF'/%0minlabe
lspacingGFgxF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%I1uvG6\",,*&)I(uinfty1GF$\"\"$\"\"\"I(vinfty1GF$F*F**&
)F(\"\"#F*I(vinfty0GF$F*!\"\"*(#F*F.F*,(*(\"\"%F*I(uinfty0GF$F*F+F*F0*
$)F+F.F*F0*&F.F*I)tinfty10GF$F*F*F*F(F*F**(F2F*,&*&F.F*I#t1GF$F*F0*&F.
F*F/F*F*F*F+F*F**(F2F*F/F*,&I#t2GF$F**&F.F*F6F*F0F*F0" }}{PARA 11 "" 1
 "" {XPPMATH 20 ">I%I2uvG6\",.*(#\"\"\"\"\")F(,&I#t2GF$!\"\"*&\"\"#F(I
(uinfty0GF$F(F,F()I(vinfty1GF$F.F(F(**F'F(,&F+F(F-F(F(,(*&F.F()I(uinft
y1GF$F.F(F(F+F(F-F,F(F1F(F(*(F'F(,&*(\"\"%F(F7F(I(vinfty0GF$F(F,*&F;F(
I)tinfty10GF$F(F(F(F/F(F(*(F'F(,&*(F.F(F7F(F<F(F,*&F.F(F>F(F,F(F+F(F(*
(#F(F.F(I#t1GF$F(F<F(F,*&#F(F;F()F<F.F(F(" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 25 "tdA1uv:=simplify(tdA1QR);" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 26 "tdA2uv:=simplify(tdA2QR);\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 31 "dtdA1uvdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 98 "f
or i from 1 to 2 do for j from 1 to 2 do dtdA1uvdlambda[i,j]:= diff(td
A1uv[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 31 "dtdA2uvdlambda:=Matrix
(2,2,0):\n" }{MPLTEXT 1 0 98 "for i from 1 to 2 do for j from 1 to 2 d
o dtdA2uvdlambda[i,j]:= diff(tdA2uv[i,j],lambda): od: od:\n" }{MPLTEXT
 1 0 16 "dtdA1uvdlambda:\n" }{MPLTEXT 1 0 16 "dtdA2uvdlambda:\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "dtdLuvdt1:=Matrix(2,2,0):\n" }
{MPLTEXT 1 0 69 "for i from 1 to 2 do for j from 1 to 2 do dtdLuvdt1[i
,j]:=simplify( \n" }{MPLTEXT 1 0 169 "diff(tdLuv[i,j],uinfty0)*duinfty
0dt1+diff(tdLuv[i,j],uinfty1)*duinfty1dt1+diff(tdLuv[i,j],vinfty0)*dvi
nfty0dt1+diff(tdLuv[i,j],vinfty1)*dvinfty1dt1 +diff(tdLuv[i,j],t1)\n" 
}{MPLTEXT 1 0 11 "): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 
"dtdLuvdt2:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 59 "for i from 1 to 2 do f
or j from 1 to 2 do dtdLuvdt2[i,j]:=\n" }{MPLTEXT 1 0 178 "diff(tdLuv[
i,j],uinfty0)*duinfty0dt2+diff(tdLuv[i,j],uinfty1)*duinfty1dt2+diff(td
Luv[i,j],vinfty0)*dvinfty0dt2+diff(tdLuv[i,j],vinfty1)*dvinfty1dt2 +di
ff(tdLuv[i,j],t2): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 107 "
CompatibilityEquation1:=simplify(dtdLuvdt1-dtdA1uvdlambda+Multiply(tdL
uv,tdA1uv)-Multiply(tdA1uv,tdLuv)): \n" }{MPLTEXT 1 0 105 "Compatibili
tyEquation2:=simplify(dtdLuvdt2-dtdA2uvdlambda+Multiply(tdLuv,tdA2uv)-
Multiply(tdA2uv,tdLuv)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6
#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$
66-I$mtrGF$6'-I$mtdGF$6(-F,6%-I#miGF$6&Q(uinfty1F'/%'italicGQ%trueF'/%
+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6.Q(&minus;F'F
@/FDQ'normalF'/%&fenceGQ&falseF'/%*separatorGFN/%)stretchyGFN/%*symmet
ricGFN/%(largeopGFN/%.movablelimitsGFN/%'accentGFN/%'lspaceGQ,0.222222
2emF'/%'rspaceGFgn-F:6&Q)&lambda;F'/F>FNF@FJ/%)rowalignGQ!F'/%,columna
lignGF`o/%+groupalignGF`o/%(rowspanGQ\"1F'/%+columnspanGFgo-F56(-I#mnG
F$6%FgoF@FJF^oFaoFcoFeoFhoF^oFaoFco-F26'-F56(-F,6%-F]p6%Q\"2F'F@FJ-FG6
-Q1&InvisibleTimes;F'FJFLFOFQFSFUFWFY/FfnQ&0.0emF'/FinF\\q-F:6&Q(vinft
y1F'F=F@FCF^oFaoFcoFeoFho-F56(-F,6&-FG6.Q*&uminus0;F'F@FJFLFOFQFSFUFWF
YFenFhnF9-FG6.Q\"+F'F@FJFLFOFQFSFUFWFYFenFhnFjnF^oFaoFcoFeoFhoF^oFaoFc
o/%&alignGQ%axisF'/F_oQ)baselineF'/FboQ'centerF'/FdoQ'|frleft|hrF'/%/a
lignmentscopeGF?/%,columnwidthGQ%autoF'/%&widthGFhr/%+rowspacingGQ&1.0
exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFcs/%
&frameGFcs/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFN/%-equalcolumn
sGFN/%-displaystyleGFN/%%sideGQ&rightF'/%0minlabelspacingGF`sF@FJ/%%op
enGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6
#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$
66-I$mtrGF$6'-I$mtdGF$6(-F,6*-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0
,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFE/%)st
retchyGFE/%*symmetricGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%
'lspaceGQ,0.2222222emF'/%'rspaceGFT-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'F
=F@-Fgn6%Q\"2F'F=F@/%.linethicknessGFin/%+denomalignGQ'centerF'/%)numa
lignGFao/%)bevelledGFEF=-F:6-Q1&InvisibleTimes;F'F@FCFFFHFJFLFNFP/FSQ&
0.0emF'/FVFjo-F,6#-I%msupGF$6%-I#miGF$6&Q)&lambda;F'/%'italicGFEF=F@Fj
n/%1superscriptshiftGQ\"0F'-F:6.Q\"+F'F=F@FCFFFHFJFLFNFPFRFU-F,6%FYFfo
-F,6#-Fbp6&Q(uinfty0F'/FfpQ%trueF'F=/FAQ'italicF'-F:6.Q(&minus;F'F=F@F
CFFFHFJFLFNFPFRFU-F,6%-FZ6)Ffn-Fgn6%Q\"4F'F=F@F]oF_oFboFdoF=Ffo-F,6#-F
bp6&Q#t2F'FdqF=FfqFjp-F,6%FYFfo-F,6#-Fbp6&Q(vinfty1F'FdqF=Ffq/%)rowali
gnGQ!F'/%,columnalignGF`s/%+groupalignGF`s/%(rowspanGFin/%+columnspanG
Fin-F56(-F,6%-F,6%FYFfo-F,6#FapFjp-F,6%FYFfo-F,6#-Fbp6&Q(uinfty1F'FdqF
=FfqF^sFasFcsFesFgsF^sFasFcs-F26'-F56(-F,6%-F,6%-F#6%-F,6&F9FetFjpFapF
=F@FfoF[sFjp-Fbp6&Q(vinfty0F'FdqF=FfqF^sFasFcsFesFgs-F56(-F,6)FWFhqF]q
FjpF[rFhqFgrF^sFasFcsFesFgsF^sFasFcs/%&alignGQ%axisF'/F_sQ)baselineF'/
FbsFao/FdsQ'|frleft|hrF'/%/alignmentscopeGFeq/%,columnwidthGQ%autoF'/%
&widthGFgv/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlines
GQ%noneF'/%,columnlinesGFbw/%&frameGFbw/%-framespacingGQ,0.4em~0.5exF'
/%*equalrowsGFE/%-equalcolumnsGFE/%-displaystyleGFE/%%sideGQ&rightF'/%
0minlabelspacingGF_wF=F@/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 218 62 "Computation of the evolutions of the Dar
boux coordinates (u,v)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "s
implify(series(CompatibilityEquation1[1,1],lambda=0)):" }{MPLTEXT 1 0 
1 "\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 47 "series(Compatibil
ityEquation1[1,2],lambda=0)):\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT
 1 0 46 "series(CompatibilityEquation1[2,1],lambda=0)):" }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 35 "series(Compatib
ilityEquation1[2,2]+" }{MPLTEXT 1 0 27 "CompatibilityEquation1[1,1]" }
{MPLTEXT 1 0 13 ",lambda=0));\n" }{MPLTEXT 1 0 43 "dvinfty1dt1:=-4*uin
fty1*vinfty1+2*vinfty0;\n" }{MPLTEXT 1 0 86 "dvinfty0dt1:=-vinfty1^2+2
*(uinfty1^2-2*uinfty0)*vinfty1-2*uinfty1*vinfty0+2*tinfty10;\n" }
{MPLTEXT 1 0 49 "duinfty1dt1:=2*uinfty1^2+t2-2*uinfty0-2*vinfty1;\n" }
{MPLTEXT 1 0 52 "duinfty0dt1:=(t2+2*uinfty0)*uinfty1+2*t1-2*vinfty0;\n
" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEquation1[1,1],lambda
=0));\n" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEquation1[1,2]
,lambda=0));\n" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEquatio
n1[2,1],lambda=0));\n" }{MPLTEXT 1 0 84 "simplify(series(Compatibility
Equation1[2,2]+CompatibilityEquation1[1,1],lambda=0));\n" }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEquation2[1,1]
,lambda=0)):\n" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEquatio
n2[1,2],lambda=0)):\n" }{MPLTEXT 1 0 56 "simplify(series(Compatibility
Equation2[2,1],lambda=0)):\n" }{MPLTEXT 1 0 84 "simplify(series(Compat
ibilityEquation2[2,2]+CompatibilityEquation2[1,1],lambda=0));\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 91 "dvinfty1dt2:=vinfty1*uinfty1^2-u
infty1*vinfty0-(1/2)*vinfty1^2-2*uinfty0*vinfty1+tinfty10;\n" }
{MPLTEXT 1 0 130 "dvinfty0dt2:=uinfty1^3*vinfty1-uinfty1^2*vinfty0-(1/
2)*(-2*t2*vinfty1+vinfty1^2-2*tinfty10)*uinfty1-(1/2)*(t2+2*uinfty0)*v
infty0;\n" }{MPLTEXT 1 0 54 "duinfty1dt2:=(1/2)*(t2+2*uinfty0)*uinfty1
+t1-vinfty0;\n" }{MPLTEXT 1 0 89 "duinfty0dt2:=(t1-vinfty0)*uinfty1-(1
/4)*t2^2+(1/2)*t2*vinfty1+uinfty0^2+uinfty0*vinfty1;\n" }{MPLTEXT 1 0 
1 "\n" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEquation2[1,1],l
ambda=0));\n" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEquation2
[1,2],lambda=0));\n" }{MPLTEXT 1 0 56 "simplify(series(CompatibilityEq
uation2[2,1],lambda=0));\n" }{MPLTEXT 1 0 83 "simplify(series(Compatib
ilityEquation2[2,2]+CompatibilityEquation2[1,1],lambda=0));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dvin
fty1dt1G6\",&*(\"\"%\"\"\"I(uinfty1GF$F(I(vinfty1GF$F(!\"\"*&\"\"#F(I(
vinfty0GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dvinfty0dt1G6\",**$
)I(vinfty1GF$\"\"#\"\"\"!\"\"*(F)F*,&*$)I(uinfty1GF$F)F*F**&F)F*I(uinf
ty0GF$F*F+F*F(F*F**(F)F*F0F*I(vinfty0GF$F*F+*&F)F*I)tinfty10GF$F*F*" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I,duinfty1dt1G6\",**&\"\"#\"\"\")I(uin
fty1GF$F'F(F(I#t2GF$F(*&F'F(I(uinfty0GF$F(!\"\"*&F'F(I(vinfty1GF$F(F."
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,duinfty0dt1G6\",(*&,&I#t2GF$\"\"\"
*&\"\"#F)I(uinfty0GF$F)F)F)I(uinfty1GF$F)F)*&F+F)I#t1GF$F)F)*&F+F)I(vi
nfty0GF$F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dvinfty1dt2G6\",,*&I(vinfty1
GF$\"\"\")I(uinfty1GF$\"\"#F(F(*&F*F(I(vinfty0GF$F(!\"\"*&#F(F+F()F'F+
F(F.*(F+F(I(uinfty0GF$F(F'F(F.I)tinfty10GF$F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I,dvinfty0dt2G6\",**&)I(uinfty1GF$\"\"$\"\"\"I(vinfty1GF
$F*F**&)F(\"\"#F*I(vinfty0GF$F*!\"\"*(#F*F.F*,(*(F.F*I#t2GF$F*F+F*F0*$
)F+F.F*F**&F.F*I)tinfty10GF$F*F0F*F(F*F0*(F2F*,&F5F**&F.F*I(uinfty0GF$
F*F*F*F/F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,duinfty1dt2G6\",(*(#
\"\"\"\"\"#F(,&I#t2GF$F(*&F)F(I(uinfty0GF$F(F(F(I(uinfty1GF$F(F(I#t1GF
$F(I(vinfty0GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,duinfty0dt2G6
\",,*&,&I#t1GF$\"\"\"I(vinfty0GF$!\"\"F)I(uinfty1GF$F)F)*&#F)\"\"%F))I
#t2GF$\"\"#F)F+*(#F)F2F)F1F)I(vinfty1GF$F)F)*$)I(uinfty0GF$F2F)F)*&F8F
)F5F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 83 "Ve
rification of the isospectral condition \\delta_t[\\td\{L\}]= \\partia
l_\\lambda \\td\{A\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dif
f(tdLuv[1,1],t1)-" }{MPLTEXT 1 0 19 "dtdA1uvdlambda[1,1]" }{MPLTEXT 1 
0 2 ";\n" }{MPLTEXT 1 0 41 "diff(tdLuv[1,2],t1)-dtdA1uvdlambda[1,2];\n
" }{MPLTEXT 1 0 41 "diff(tdLuv[2,1],t1)-dtdA1uvdlambda[2,1];\n" }
{MPLTEXT 1 0 41 "diff(tdLuv[2,2],t1)-dtdA1uvdlambda[2,2];\n" }{MPLTEXT
 1 0 1 "\n" }{MPLTEXT 1 0 41 "diff(tdLuv[1,1],t2)-dtdA2uvdlambda[1,1];
\n" }{MPLTEXT 1 0 41 "diff(tdLuv[1,2],t2)-dtdA2uvdlambda[1,2];\n" }
{MPLTEXT 1 0 41 "diff(tdLuv[2,1],t2)-dtdA2uvdlambda[2,1];\n" }{MPLTEXT
 1 0 40 "diff(tdLuv[2,2],t2)-dtdA2uvdlambda[2,2];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 218 228 "Final c
hange of coordinates to obtain a canonical set of isospectral coordina
tes (x_2,x_3,y_2,y_3) inspired by Appendix A of \"Hamiltonian structur
e of rational isomonodromic deformation systems\" by Bertola, Harnad a
nd Hurtubise." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 48 "Definition of t
he change od Darboux coordinates." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "y2sol:=alpha*vinfty1;\n" }{MPLTEXT 1 0 22 "x3sol:=-be
ta*uinfty1;\n" }{MPLTEXT 1 0 38 "x2sol:=delta*uinfty0+epsilon*vinfty1;
\n" }{MPLTEXT 1 0 37 "y3sol:=mu*vinfty0-nu*uinfty1*vinfty1;" }{MPLTEXT
 1 0 1 "\n" }{MPLTEXT 1 0 10 "solve(\{y2=" }{MPLTEXT 1 0 9 "y2sol,y3="
 }{MPLTEXT 1 0 10 "y3sol,x2=x" }{MPLTEXT 1 0 50 "2sol,x3=x3sol\},\{uin
fty0,uinfty1,vinfty0,vinfty1\});" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 
50 "uinfty0sol :=(alpha*x2-epsilon*y2)/(alpha*delta);\n" }{MPLTEXT 1 
0 23 "uinfty1sol :=-x3/beta;\n" }{MPLTEXT 1 0 55 "vinfty0sol :=(alpha*
beta*y3-nu*x3*y2)/(alpha*beta*mu);\n" }{MPLTEXT 1 0 23 "vinfty1sol :=y
2/alpha;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 66 "dy2dt1function:=una
pply( simplify(diff(y2sol,uinfty0)*duinfty0dt1+" }{MPLTEXT 1 0 32 "dif
f(y2sol,uinfty1)*duinfty1dt1+" }{MPLTEXT 1 0 32 "diff(y2sol,vinfty0)*d
vinfty0dt1+" }{MPLTEXT 1 0 68 "diff(y2sol,vinfty1)*dvinfty1dt1), uinft
y0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 66 "dy3dt1function:=unap
ply( simplify(diff(y3sol,uinfty0)*duinfty0dt1+" }{MPLTEXT 1 0 132 "dif
f(y3sol,uinfty1)*duinfty1dt1+diff(y3sol,vinfty0)*dvinfty0dt1+diff(y3so
l,vinfty1)*dvinfty1dt1), uinfty0,uinfty1,vinfty0,vinfty1):\n" }
{MPLTEXT 1 0 66 "dx2dt1function:=unapply( simplify(diff(x2sol,uinfty0)
*duinfty0dt1+" }{MPLTEXT 1 0 132 "diff(x2sol,uinfty1)*duinfty1dt1+diff
(x2sol,vinfty0)*dvinfty0dt1+diff(x2sol,vinfty1)*dvinfty1dt1), uinfty0,
uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 198 "dx3dt1function:=unappl
y( simplify(diff(x3sol,uinfty0)*duinfty0dt1+diff(x3sol,uinfty1)*duinft
y1dt1+diff(x3sol,vinfty0)*dvinfty0dt1+diff(x3sol,vinfty1)*dvinfty1dt1)
, uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 
1 0 17 "dy2dt1:=simplify(" }{MPLTEXT 1 0 15 "dy2dt1function(" }
{MPLTEXT 1 0 11 "uinfty0sol," }{MPLTEXT 1 0 22 "uinfty1sol,vinfty0sol,
" }{MPLTEXT 1 0 14 "vinfty1sol)):\n" }{MPLTEXT 1 0 79 "dy3dt1:=simplif
y(dy3dt1function(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }
{MPLTEXT 1 0 79 "dx2dt1:=simplify(dx2dt1function(uinfty0sol,uinfty1sol
,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 1 0 79 "dx3dt1:=simplify(dx3dt1
function(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 
1 0 1 "\n" }{MPLTEXT 1 0 66 "dy2dt2function:=unapply( simplify(diff(y2
sol,uinfty0)*duinfty0dt2+" }{MPLTEXT 1 0 132 "diff(y2sol,uinfty1)*duin
fty1dt2+diff(y2sol,vinfty0)*dvinfty0dt2+diff(y2sol,vinfty1)*dvinfty1dt
2), uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 198 "dy3dt2func
tion:=unapply( simplify(diff(y3sol,uinfty0)*duinfty0dt2+diff(y3sol,uin
fty1)*duinfty1dt2+diff(y3sol,vinfty0)*dvinfty0dt2+diff(y3sol,vinfty1)*
dvinfty1dt2), uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 198 "
dx2dt2function:=unapply( simplify(diff(x2sol,uinfty0)*duinfty0dt2+diff
(x2sol,uinfty1)*duinfty1dt2+diff(x2sol,vinfty0)*dvinfty0dt2+diff(x2sol
,vinfty1)*dvinfty1dt2), uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT
 1 0 198 "dx3dt2function:=unapply( simplify(diff(x3sol,uinfty0)*duinft
y0dt2+diff(x3sol,uinfty1)*duinfty1dt2+diff(x3sol,vinfty0)*dvinfty0dt2+
diff(x3sol,vinfty1)*dvinfty1dt2), uinfty0,uinfty1,vinfty0,vinfty1):\n"
 }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 79 "dy2dt2:=simplify(dy2dt2functio
n(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 1 0 79 "
dy3dt2:=simplify(dy3dt2function(uinfty0sol,uinfty1sol,vinfty0sol,vinft
y1sol)):\n" }{MPLTEXT 1 0 79 "dx2dt2:=simplify(dx2dt2function(uinfty0s
ol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 1 0 78 "dx3dt2:=si
mplify(dx3dt2function(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I&y2solG6\"*&I&alphaGF$\"\"\"I(vinfty
1GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&x3solG6\",$*&I%betaGF$\"\"
\"I(uinfty1GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&x2solG6\",&*
&I&deltaGF$\"\"\"I(uinfty0GF$F(F(*&I(epsilonGF$F(I(vinfty1GF$F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I&y3solG6\",&*(I#nuGF$\"\"\"I(uinfty1GF
$F(I(vinfty1GF$F(!\"\"*&I#muGF$F(I(vinfty0GF$F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<&/I(uinfty0G6\"*(,&*&I&alphaGF%\"\"\"I#x2GF%F*F**&I(epsi
lonGF%F*I#y2GF%F*!\"\"F*F)F/I&deltaGF%F//I(uinfty1GF%,$*&I#x3GF%F*I%be
taGF%F/F//I(vinfty0GF%**,&*(F)F*F6F*I#y3GF%F*F**(I#nuGF%F*F5F*F.F*F/F*
F)F/F6F/I#muGF%F//I(vinfty1GF%*&F.F*F)F/" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I+uinfty0solG6\"*(,&*&I&alphaGF$\"\"\"I#x2GF$F)F)*&I(epsilonGF$F)
I#y2GF$F)!\"\"F)F(F.I&deltaGF$F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+u
infty1solG6\",$*&I#x3GF$\"\"\"I%betaGF$!\"\"F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I+vinfty0solG6\"**,&*(I&alphaGF$\"\"\"I%betaGF$F)I#y3GF$
F)F)*(I#nuGF$F)I#x3GF$F)I#y2GF$F)!\"\"F)F(F0F*F0I#muGF$F0" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">I+vinfty1solG6\"*&I#y2GF$\"\"\"I&alphaGF$!\"\"" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 61 "Expression of the Hamiltonians
 in the coordinates (x_i,y_i). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "mu:=1;\n" }{MPLTEXT 1 0 10 "beta:=-1;\n" }{MPLTEXT 1 0 12 "epsi
lon:=0;\n" }{MPLTEXT 1 0 10 "delta:=1;\n" }{MPLTEXT 1 0 8 "nu:=mu;\n" 
}{MPLTEXT 1 0 22 "alpha:=-beta*mu/delta;" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 220 "Ham1xy:=((((-delta*t2-2*x2)*x3+
2*delta*t1*beta)*mu-2*y3*beta*(delta-epsilon))*alpha*y2+3*y2^2*(mu*eps
ilon+(1/3)*nu*(delta-epsilon))*x3)/(alpha*beta*mu)-2*y3*alpha*x2/mu-(2
*beta*mu*x3*tinfty10+2*x3^2*y3)/beta-beta*t2*y3:\n" }{MPLTEXT 1 0 10 "
simplify( " }{MPLTEXT 1 0 25 "dx2dt1-diff(Ham1xy,y2));\n" }{MPLTEXT 1 
0 16 "simplify(series(" }{MPLTEXT 1 0 10 "simplify( " }{MPLTEXT 1 0 
30 "dy2dt1+diff(Ham1xy,x2)),y2));\n" }{MPLTEXT 1 0 9 "simplify(" }
{MPLTEXT 1 0 7 "series(" }{MPLTEXT 1 0 40 "simplify( dy3dt1+diff(Ham1x
y,x3)),y2));\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 47 "series(s
implify( dx3dt1-diff(Ham1xy,y3)),y2));\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 148 "Ham2xy:=((-(1/2)*x3^2+(1/4)*t2+
(1/2)*x2)*y2+x3*(t1-y3)-(1/4)*t2^2+x2^2)*y2+(x3*y3-tinfty10)*x2+(1/2)*
t2*y3*x3+(1/2)*y3*(2*t1-y3)+(1/2)*tinfty10*t2:\n" }{MPLTEXT 1 0 10 "si
mplify( " }{MPLTEXT 1 0 25 "dx2dt2-diff(Ham2xy,y2));\n" }{MPLTEXT 1 0 
56 "simplify(series(simplify( dy2dt2+diff(Ham2xy,x2)),y2));\n" }
{MPLTEXT 1 0 56 "simplify(series(simplify( dy3dt2+diff(Ham2xy,x3)),y2)
);\n" }{MPLTEXT 1 0 55 "simplify(series(simplify( dx3dt2-diff(Ham2xy,y
3)),y2));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#muG6\"\"\"\"" }}{PARA 11
 "" 1 "" {XPPMATH 20 ">I%betaG6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I(epsilonG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&deltaG6\"\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#nuG6\"\"\"\"" }}{PARA 11 "" 1
 "" {XPPMATH 20 ">I&alphaG6\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Ham1xy:=simplify(H
am1xy);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "x2sol:=x2sol;\n" }
{MPLTEXT 1 0 14 "x3sol:=x3sol;\n" }{MPLTEXT 1 0 14 "y2sol:=y2sol;\n" }
{MPLTEXT 1 0 13 "y3sol:=y3sol;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 
"Ham2xy:=expand(simplify(Ham2xy));" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I'Ham1xyG6\",**(\"\"#\"\"\")I#x3GF$F'F(I#y3GF$F(F(
*&,(*$)I#y2GF$F'F(!\"\"*&,&I#t2GF$F(*&F'F(I#x2GF$F(F(F(F0F(F(*&F'F(I)t
infty10GF$F(F1F(F*F(F(*&,&*&F'F(I#t1GF$F(F(*&F'F(F+F(F1F(F0F(F(*&F+F(,
&F4F(F5F1F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&x2solG6\"I(uinfty0GF
$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&x3solG6\"I(uinfty1GF$" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I&y2solG6\"I(vinfty1GF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I&y3solG6\",&*&I(uinfty1GF$\"\"\"I(vinfty1GF$F(!\"\"I(vi
nfty0GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Ham2xyG6\",<*(#\"\"\"\"
\"#F()I#x3GF$F)F()I#y2GF$F)F(!\"\"*(#F(\"\"%F(F,F(I#t2GF$F(F(*(F'F(I#x
2GF$F(F,F(F(*(F-F(F+F(I#t1GF$F(F(*(F-F(F+F(I#y3GF$F(F.*(F0F(F-F()F2F)F
(F.*&F-F()F4F)F(F(*(F4F(F+F(F8F(F(*&F4F(I)tinfty10GF$F(F.**F'F(F2F(F8F
(F+F(F(*&F8F(F6F(F(*&F'F()F8F)F(F.*(F'F(F?F(F2F(F(" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 218 115 "Expression of the spectral invariants in the co
ordinates (x,y) and verification that they recover the Hamiltonians." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "I1uvfunction:=unapply(I1u
v,uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 15 "I1xy:=simplif
y(" }{MPLTEXT 1 0 13 "I1uvfunction(" }{MPLTEXT 1 0 47 "uinfty0sol,uinf
ty1sol,vinfty0sol,vinfty1sol));\n" }{MPLTEXT 1 0 61 "I2uvfunction:=una
pply(I2uv,uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 74 "I2xy:
=simplify(I2uvfunction(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol));"
 }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 28 "simplify(Ham1xy-(-2)*I1xy);\n
" }{MPLTEXT 1 0 27 "simplify(Ham2xy-(-2)*I2xy);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%I1xyG6\",**&)I#x3GF$\"\"#\"\"\"I#y3GF$F*!\"\"*(#F*F)F*
,(*$)I#y2GF$F)F*F**&,&I#t2GF$F,*&F)F*I#x2GF$F*F,F*F2F*F**&F)F*I)tinfty
10GF$F*F*F*F(F*F**(F.F*,&*&F)F*I#t1GF$F*F,*&F)F*F+F*F*F*F2F*F**(F.F*F+
F*,&F5F*F6F,F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%I2xyG6\",.*(#\"\"
\"\"\")F(,(*&\"\"#F()I#x3GF$F,F(F(I#t2GF$!\"\"*&F,F(I#x2GF$F(F0F()I#y2
GF$F,F(F(*(F'F(,(*&,&*&\"\"%F(I#t1GF$F(F0*&F:F(I#y3GF$F(F(F(F.F(F(*$)F
/F,F(F(*&F:F()F2F,F(F0F(F4F(F(**#F(F:F(F=F(,&F/F(F1F(F(F.F(F0*&FCF()F=
F,F(F(*(#F(F,F(F=F(F;F(F0*(FCF(I)tinfty10GF$F(,&F/F(F1F0F(F0" }}{PARA 
11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT 218 55 "Expression of the Lax matrices
 in the coordinates (x,y)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "tdLxy11function:=unapply(tdLuv[1,1]," }{MPLTEXT 1 0 34 "uinfty0,ui
nfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 36 "tdLxy12function:=unapply(
tdLuv[1,2]," }{MPLTEXT 1 0 34 "uinfty0,uinfty1,vinfty0,vinfty1):\n" }
{MPLTEXT 1 0 36 "tdLxy21function:=unapply(tdLuv[2,1]," }{MPLTEXT 1 0 
34 "uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 36 "tdLxy22func
tion:=unapply(tdLuv[2,2]," }{MPLTEXT 1 0 33 "uinfty0,uinfty1,vinfty0,v
infty1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "tdLxy:=Matrix(2,2,0):
\n" }{MPLTEXT 1 0 21 "tdLxy[1,1]:=simplify(" }{MPLTEXT 1 0 16 "tdLxy11
function(" }{MPLTEXT 1 0 43 "uinfty0sol,uinfty1sol,vinfty0sol,vinfty1s
ol" }{MPLTEXT 1 0 4 ")): " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 21 "tdLx
y[1,2]:=simplify(" }{MPLTEXT 1 0 59 "tdLxy12function(uinfty0sol,uinfty
1sol,vinfty0sol,vinfty1sol" }{MPLTEXT 1 0 4 ")):\n" }{MPLTEXT 1 0 21 "
tdLxy[2,1]:=simplify(" }{MPLTEXT 1 0 59 "tdLxy21function(uinfty0sol,ui
nfty1sol,vinfty0sol,vinfty1sol" }{MPLTEXT 1 0 3 ")):" }{MPLTEXT 1 0 1 
"\n" }{MPLTEXT 1 0 21 "tdLxy[2,2]:=simplify(" }{MPLTEXT 1 0 59 "tdLxy2
2function(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol" }{MPLTEXT 1 0 
3 ")):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 6 "tdLxy;" }{MPLTEXT 1 0 1 
"\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 72 "tdA1x
y11function:=unapply(tdA1uv[1,1],uinfty0,uinfty1,vinfty0,vinfty1):\n" 
}{MPLTEXT 1 0 72 "tdA1xy12function:=unapply(tdA1uv[1,2],uinfty0,uinfty
1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 72 "tdA1xy21function:=unapply(tdA
1uv[2,1],uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 72 "tdA1xy
22function:=unapply(tdA1uv[2,2],uinfty0,uinfty1,vinfty0,vinfty1):\n" }
{MPLTEXT 1 0 23 "tdA1xy:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "tdA1xy[1,
1]:=simplify(tdA1xy11function(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1
sol)): \n" }{MPLTEXT 1 0 86 "tdA1xy[1,2]:=simplify(tdA1xy12function(ui
nfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 1 0 86 "tdA1
xy[2,1]:=simplify(tdA1xy21function(uinfty0sol,uinfty1sol,vinfty0sol,vi
nfty1sol)):\n" }{MPLTEXT 1 0 86 "tdA1xy[2,2]:=simplify(tdA1xy22functio
n(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 1 0 7 "t
dA1xy;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 72 "td
A2xy11function:=unapply(tdA2uv[1,1],uinfty0,uinfty1,vinfty0,vinfty1):
\n" }{MPLTEXT 1 0 72 "tdA2xy12function:=unapply(tdA2uv[1,2],uinfty0,ui
nfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 72 "tdA2xy21function:=unapply
(tdA2uv[2,1],uinfty0,uinfty1,vinfty0,vinfty1):\n" }{MPLTEXT 1 0 72 "td
A2xy22function:=unapply(tdA2uv[2,2],uinfty0,uinfty1,vinfty0,vinfty1):
\n" }{MPLTEXT 1 0 23 "tdA2xy:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "tdA2
xy[1,1]:=simplify(tdA2xy11function(uinfty0sol,uinfty1sol,vinfty0sol,vi
nfty1sol)): \n" }{MPLTEXT 1 0 86 "tdA2xy[1,2]:=simplify(tdA2xy12functi
on(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 1 0 86 
"tdA2xy[2,1]:=simplify(tdA2xy21function(uinfty0sol,uinfty1sol,vinfty0s
ol,vinfty1sol)):\n" }{MPLTEXT 1 0 86 "tdA2xy[2,2]:=simplify(tdA2xy22fu
nction(uinfty0sol,uinfty1sol,vinfty0sol,vinfty1sol)):\n" }{MPLTEXT 1 
0 7 "tdA2xy;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulena
meG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6
'-I$mtdGF$6(-F,6,-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,ma
thvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*
symmetricGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.
2222222emF'/%'rspaceGFT-F,6#-I%msupGF$6%-I#miGF$6&Q)&lambda;F'/%'itali
cGFEF=F@-I#mnGF$6%Q\"3F'F=F@/%1superscriptshiftGQ\"0F'-F:6.Q\"+F'F=F@F
CFFFHFJFLFNFPFRFU-F,6%-F#6%-F,6&F9-Fgn6&Q#t2F'/F[oQ%trueF'F=/FAQ'itali
cF'Fco-Fgn6&Q#y2F'F_pF=FapF=F@-F:6-Q1&InvisibleTimes;F'F@FCFFFHFJFLFNF
P/FSQ&0.0emF'/FVFjpFfn-F:6.Q(&minus;F'F=F@FCFFFHFJFLFNFPFRFU-Fgn6&Q#t1
F'F_pF=FapFco-F,6%-Fgn6&Q#x3F'F_pF=FapFfpFcpFco-Fgn6&Q#y3F'F_pF=Fap/%)
rowalignGQ!F'/%,columnalignGF\\r/%+groupalignGF\\r/%(rowspanGQ\"1F'/%+
columnspanGFcr-F56(-F,6)-F,6%FdqFfpFfnFco-F,6#-FZ6%Ffn-F]o6%Q\"2F'F=F@
F`oFco-Fgn6&Q#x2F'F_pF=FapFco-F,6%-I&mfracGF$6)-F]o6%FcrF=F@F`s/%.line
thicknessGFcr/%+denomalignGQ'centerF'/%)numalignGFat/%)bevelledGFEF=Ff
p-F,6#F\\pFjqF]rF_rFarFdrFjqF]rF_r-F26'-F56(-F,6,F9-F,6#-FZ6%FcpF`sF`o
Fco-F,6%-F#6%-F,6'-F,6%F`sFfpF^sFcoF\\pF\\q-F,6%F`sFfpFcsF=F@FfpFcpF
\\q-F,6'F`sFfpFdqFfpFgqFco-F,6'F`sFfpFfnFfpFgqFco-F,6%F`sFfp-Fgn6&Q)ti
nfty10F'F_pF=FapFjqF]rF_rFarFdr-F56(-F,6+FWFco-F,6%-F#6%-F,6%F\\pF\\qF
cpF=F@FfpFfnFcoF_qF\\qFbqF\\qFgqFjqF]rF_rFarFdrFjqF]rF_r/%&alignGQ%axi
sF'/F[rQ)baselineF'/F^rFat/F`rQ'|frleft|hrF'/%/alignmentscopeGF`p/%,co
lumnwidthGQ%autoF'/%&widthGF[x/%+rowspacingGQ&1.0exF'/%.columnspacingG
Q&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFfx/%&frameGFfx/%-framespa
cingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-equalcolumnsGFE/%-displaystyleGF
E/%%sideGQ&rightF'/%0minlabelspacingGFcxF=F@/%%openGQ\"[F'/%&closeGQ\"
]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,T
ypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF
$6(-F,6%-I#miGF$6&Q#x3F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,
mathvariantGQ'italicF'-I#moGF$6.Q(&minus;F'F@/FDQ'normalF'/%&fenceGQ&f
alseF'/%*separatorGFN/%)stretchyGFN/%*symmetricGFN/%(largeopGFN/%.mova
blelimitsGFN/%'accentGFN/%'lspaceGQ,0.2222222emF'/%'rspaceGFgn-F:6&Q)&
lambda;F'/F>FNF@FJ/%)rowalignGQ!F'/%,columnalignGF`o/%+groupalignGF`o/
%(rowspanGQ\"1F'/%+columnspanGFgo-F56(-I#mnGF$6%FgoF@FJF^oFaoFcoFeoFho
F^oFaoFco-F26'-F56(-F,6%-F]p6%Q\"2F'F@FJ-FG6-Q1&InvisibleTimes;F'FJFLF
OFQFSFUFWFY/FfnQ&0.0emF'/FinF\\q-F:6&Q#y2F'F=F@FCF^oFaoFcoFeoFho-F56(-
F,6%FjnFFF9F^oFaoFcoFeoFhoF^oFaoFco/%&alignGQ%axisF'/F_oQ)baselineF'/F
boQ'centerF'/FdoQ'|frleft|hrF'/%/alignmentscopeGF?/%,columnwidthGQ%aut
oF'/%&widthGFbr/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)row
linesGQ%noneF'/%,columnlinesGF]s/%&frameGF]s/%-framespacingGQ,0.4em~0.
5exF'/%*equalrowsGFN/%-equalcolumnsGFN/%-displaystyleGFN/%%sideGQ&righ
tF'/%0minlabelspacingGFjrF@FJ/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11
 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_s
yslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6*-I#moGF
$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&f
enceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGF
E/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222emF'/%'rspaceGFT-
F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'F=F@-Fgn6%Q\"2F'F=F@/%.linethicknessG
Fin/%+denomalignGQ'centerF'/%)numalignGFao/%)bevelledGFEF=-F:6-Q1&Invi
sibleTimes;F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFjo-F,6#-I%msupGF$6%-I#miG
F$6&Q)&lambda;F'/%'italicGFEF=F@Fjn/%1superscriptshiftGQ\"0F'-F:6.Q\"+
F'F=F@FCFFFHFJFLFNFPFRFU-F,6%FYFfo-F,6#-Fbp6&Q#x2F'/FfpQ%trueF'F=/FAQ'
italicF'-F:6.Q(&minus;F'F=F@FCFFFHFJFLFNFPFRFU-F,6%-FZ6)Ffn-Fgn6%Q\"4F
'F=F@F]oF_oFboFdoF=Ffo-F,6#-Fbp6&Q#t2F'FdqF=FfqFjp-F,6%FYFfo-F,6#-Fbp6
&Q#y2F'FdqF=Ffq/%)rowalignGQ!F'/%,columnalignGF`s/%+groupalignGF`s/%(r
owspanGFin/%+columnspanGFin-F56(-F,6%-F,6%FYFfo-F,6#FapFjp-F,6%FYFfo-F
,6#-Fbp6&Q#x3F'FdqF=FfqF^sFasFcsFesFgsF^sFasFcs-F26'-F56(-F,6%-F,6%Fap
FfoF[sFjp-Fbp6&Q#y3F'FdqF=FfqF^sFasFcsFesFgs-F56(-F,6)FWFhqF]qFjpF[rFh
qFgrF^sFasFcsFesFgsF^sFasFcs/%&alignGQ%axisF'/F_sQ)baselineF'/FbsFao/F
dsQ'|frleft|hrF'/%/alignmentscopeGFeq/%,columnwidthGQ%autoF'/%&widthGF
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