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{SECT 0 {EXCHG {PARA 216 "" 0 "" {TEXT 210 149 "In this Maple file, w
e compute the Casimir coefficients of the Lax matrix L associated to t
he Painlev\351 6 equation in relation with the spectral curve" }{TEXT 
210 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }
{MPLTEXT 1 0 38 "P1:=x-> P011/x+P111/(x-1)+Pt11/(x-t);\n" }{MPLTEXT 1 
0 73 "P2:=x-> P022/x^2+P012/x+P122/(x-1)^2+P112/(x-1)+Pt22/(x-t)^2+Pt1
2/(x-t);\n" }{MPLTEXT 1 0 45 "SpectralCurve:=unapply( y^2-P1(x)*y+P2(x
),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)opera
torGF$I&arrowGF$F$,(*&I%P011GF$\"\"\"9$!\"\"F.*&I%P111GF$F.,&F/F.F0F.F
0F.*&I%Pt11GF$F.,&F/F.I\"tGF$F0F0F.F$F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,.*&I%P0
22GF$\"\"\"9$!\"#F.*&I%P012GF$F.F/!\"\"F.*&I%P122GF$F.,&F/F.F3F.F0F.*&
I%P112GF$F.F6F3F.*&I%Pt22GF$F.,&F/F.I\"tGF$F3F0F.*&I%Pt12GF$F.F;F3F.F$
F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.SpectralCurveG6\"f*6#I\"yGF$F
$6$I)operatorGF$I&arrowGF$F$,2*$9$\"\"#\"\"\"*&,(*&I%P011GF$F/I\"xGF$!
\"\"F/*&I%P111GF$F/,&F4F/F5F/F5F/*&I%Pt11GF$F/,&F4F/I\"tGF$F5F5F/F/F-F
/F5*&I%P022GF$F/F4!\"#F/*&I%P012GF$F/F4F5F/*&I%P122GF$F/F8F?F/*&I%P112
GF$F/F8F5F/*&I%Pt22GF$F/F;F?F/*&I%Pt12GF$F/F;F5F/F$F$F$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "DiaginftySheet1:=-tinfty10/x+Unknow
n/x^2;\n" }{MPLTEXT 1 0 43 "DiaginftySheet2:=-tinfty20/x+Unknown2/x^2;
\n" }{MPLTEXT 1 0 30 "Diag0Sheet1:=t010/x+Unknown3;\n" }{MPLTEXT 1 0 
30 "Diag0Sheet2:=t020/x+Unknown4;\n" }{MPLTEXT 1 0 34 "Diag1Sheet1:=t1
10/(x-1)+Unknown5;\n" }{MPLTEXT 1 0 34 "Diag1Sheet2:=t120/(x-1)+Unknow
n6;\n" }{MPLTEXT 1 0 34 "DiagtSheet1:=tt10/(x-t)+Unknown5;\n" }
{MPLTEXT 1 0 35 "DiagtSheet2:=tt20/(x-t)+Unknown6;  " }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I0DiaginftySheet1G6\",&*&I)tinfty10GF$\"\"\"I\"xGF$!
\"\"F**&I(UnknownGF$F(F)!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0Dia
ginftySheet2G6\",&*&I)tinfty20GF$\"\"\"I\"xGF$!\"\"F**&I)Unknown2GF$F(
F)!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,Diag0Sheet1G6\",&*&I%t010
GF$\"\"\"I\"xGF$!\"\"F(I)Unknown3GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I,Diag0Sheet2G6\",&*&I%t020GF$\"\"\"I\"xGF$!\"\"F(I)Unknown4GF$F(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 ">I,Diag1Sheet1G6\",&*&I%t110GF$\"\"\",
&I\"xGF$F(!\"\"F(F+F(I)Unknown5GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I,Diag1Sheet2G6\",&*&I%t120GF$\"\"\",&I\"xGF$F(!\"\"F(F+F(I)Unknown6G
F$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,DiagtSheet1G6\",&*&I%tt10GF$
\"\"\",&I\"xGF$F(I\"tGF$!\"\"F,F(I)Unknown5GF$F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I,DiagtSheet2G6\",&*&I%tt20GF$\"\"\",&I\"xGF$F(I\"tGF$!
\"\"F,F(I)Unknown6GF$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 67 "Expre
ssion of P_1 in terms of the diagonal expansion in both sheets" }{TEXT
 216 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 17 "Study at infinity" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "series(DiaginftySheet1+Dia
ginftySheet2-P1(x),x=infinity);\n" }{MPLTEXT 1 0 45 "series(Diag0Sheet
1+Diag0Sheet2-P1(x),x=0,5);\n" }{MPLTEXT 1 0 45 "series(Diag1Sheet1+Di
ag1Sheet2-P1(x),x=1,5);\n" }{MPLTEXT 1 0 45 "series(DiagtSheet1+DiagtS
heet2-P1(x),x=t,5);\n" }{MPLTEXT 1 0 17 "P011:=t010+t020;\n" }{MPLTEXT
 1 0 17 "P111:=t110+t120;\n" }{MPLTEXT 1 0 17 "Pt11:=tt10+tt20;\n" }
{MPLTEXT 1 0 59 "CoherenceEquation1:=residue(DiaginftySheet1+Diaginfty
Sheet2" }{MPLTEXT 1 0 19 "-P1(x),x=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",.*&,,I)tinfty10G6\"!\"\"I)tinfty20GF&F'I%P011GF&F'I%P111
GF&F'I%Pt11GF&F'\"\"\"I\"xGF&F'F,*&,**&F+F,I\"tGF&F,F'F*F'I(UnknownGF&
F,I)Unknown2GF&F,F,F-!\"#F,*&,&*&F+F,F1\"\"#F'F*F'F,F-!\"$F,*&,&*&F+F,
F1\"\"$F'F*F'F,F-!\"%F,*&,&*&F+F,F1\"\"%F'F*F'F,F-!\"&F,-I\"OG6$%*prot
ectedGI(_syslibGF&6#*$F-!\"'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "+1I\"x
G6\",(I%t010GF$\"\"\"I%t020GF$F'I%P011GF$!\"\"F*,*I)Unknown3GF$F'I)Unk
nown4GF$F'I%P111GF$F'*&I%Pt11GF$F'I\"tGF$F*F'\"\"!,&F.F'*&F0F'F1!\"#F'
F',&F.F'*&F0F'F1!\"$F'\"\"#,&F.F'*&F0F'F1!\"%F'\"\"$,&F.F'*&F0F'F1!\"&
F'\"\"%-I\"OG6$%*protectedGI(_syslibGF$6#F'\"\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "+1,&I\"xG6\"\"\"\"!\"\"F&,(I%t110GF%F&I%t120GF%F&I%P111GF
%F'F',*I)Unknown5GF%F&I)Unknown6GF%F&I%P011GF%F'*&I%Pt11GF%F&,&F&F&I\"
tGF%F'F'F'\"\"!,&F/F&*(F1F&,&F'F&F3F&F'F2F'F'F&,&F/F'*(F1F&F7!\"#F2F'F
'\"\"#,&F/F&*(F1F&F7!\"$F2F'F'\"\"$,&F/F'*(F1F&F7!\"%F2F'F'\"\"%-I\"OG
6$%*protectedGI(_syslibGF%6#F&\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "+
1,&I\"xG6\"\"\"\"I\"tGF%!\"\",(I%tt10GF%F&I%tt20GF%F&I%Pt11GF%F(F(,*I)
Unknown5GF%F&I)Unknown6GF%F&*&I%P011GF%F&F'F(F(*&I%P111GF%F&,&F(F&F'F&
F(F(\"\"!,&*&F1F&F'!\"#F&*&F3F&F4F8F&F&,&*&F1F&F'!\"$F(*&F3F&F4F<F(\"
\"#,&*&F1F&F'!\"%F&*&F3F&F4FAF&\"\"$,&*&F1F&F'!\"&F(*&F3F&F4FFF(\"\"%-
I\"OG6$%*protectedGI(_syslibGF%6#F&\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I%P011G6\",&I%t010GF$\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%P111G6\",&I%t110GF$\"\"\"I%t120GF$F'" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I%Pt11G6\",&I%tt10GF$\"\"\"I%tt20GF$F'" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I3CoherenceEquation1G6\",2I)tinfty10GF$\"\"\"I)tin
fty20GF$F'I%t010GF$F'I%t020GF$F'I%t110GF$F'I%t120GF$F'I%tt10GF$F'I%tt2
0GF$F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 10 "Study at 0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "factor(series(SpectralCurve(Diag0Sh
eet1),x=0));\n" }{MPLTEXT 1 0 48 "factor(series(SpectralCurve(Diag0She
et2),x=0));\n" }{MPLTEXT 1 0 51 "EQ01:=residue(x^3*SpectralCurve(Diag0
Sheet1),x=0);\n" }{MPLTEXT 1 0 51 "EQ02:=residue(x^3*SpectralCurve(Dia
g0Sheet2),x=0);\n" }{MPLTEXT 1 0 51 "EQ03:=residue(x^2*SpectralCurve(D
iag0Sheet1),x=0);\n" }{MPLTEXT 1 0 51 "EQ04:=residue(x^2*SpectralCurve
(Diag0Sheet2),x=0);\n" }{MPLTEXT 1 0 49 "EQ05:=residue(x*SpectralCurve
(Diag0Sheet1),x=0);\n" }{MPLTEXT 1 0 49 "EQ06:=residue(x*SpectralCurve
(Diag0Sheet2),x=0);\n" }{MPLTEXT 1 0 51 "EQ07:=residue(x^0*SpectralCur
ve(Diag0Sheet1),x=0):\n" }{MPLTEXT 1 0 51 "EQ08:=residue(x^0*SpectralC
urve(Diag0Sheet2),x=0):\n" }{MPLTEXT 1 0 24 "P022:=solve(EQ05,P022);\n
" }{MPLTEXT 1 0 16 "simplify(EQ05);\n" }{MPLTEXT 1 0 16 "simplify(EQ06
);\n" }{MPLTEXT 1 0 16 "simplify(EQ07);\n" }{MPLTEXT 1 0 15 "simplify(
EQ08);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "+3I\"xG6\",&*&I%t010GF$\"\"\"I%t020GF
$F(!\"\"I%P022GF$F(!\"#*&,0*(I\"tGF$F(F'F(I%t110GF$F(F(*(F0F(F'F(I%t12
0GF$F(F(*(F0F(F'F(I)Unknown3GF$F(F(*(F0F(F)F(F5F(F**&I%P012GF$F(F0F(F(
*&F'F(I%tt10GF$F(F(*&F'F(I%tt20GF$F(F(F(F0F*F*,$*&,<*(F0\"\"#F'F(F1F(F
**(F0FAF'F(F3F(F**(F0FAF1F(F5F(F**(F0FAF3F(F5F(F**&F0FAF5FAF**&I%P112G
F$F(F0FAF(*&I%P122GF$F(F0FAF**(F0F(F:F(F5F(F**(F0F(F<F(F5F(F**&I%Pt12G
F$F(F0F(F(F9F*F;F*I%Pt22GF$F*F(F0F,F*\"\"!,$*&,:*(F0\"\"$F'F(F1F(F**(F
0FTF'F(F3F(F**(F0FTF1F(F5F(F**(F0FTF3F(F5F(F**&FGF(F0FTF(*&FIF(F0FTF,F
JF*FKF*FLF(F9F*F;F*FNF,F(F0!\"$F*F(,$*&,:*(F0\"\"%F'F(F1F(F**(F0FinF'F
(F3F(F**(F0FinF1F(F5F(F**(F0FinF3F(F5F(F**&FGF(F0FinF(*&FIF(F0FinFZFJF
*FKF*FLF(F9F*F;F*FNFZF(F0!\"%F*FA,$*&,:*(F0\"\"&F'F(F1F(F**(F0FdoF'F(F
3F(F**(F0FdoF1F(F5F(F**(F0FdoF3F(F5F(F**&FGF(F0FdoF(*&FIF(F0FdoF_oFJF*
FKF*FLF(F9F*F;F*FNF_oF(F0!\"&F*FT,$*&,:*(F0\"\"'F'F(F1F(F**(F0F_pF'F(F
3F(F**(F0F_pF1F(F5F(F**(F0F_pF3F(F5F(F**&FGF(F0F_pF(*&FIF(F0F_pFjoFJF*
FKF*FLF(F9F*F;F*FNFjoF(F0!\"'F*Fin-I\"OG6$%*protectedGI(_syslibGF$6#F(
Fdo" }}{PARA 11 "" 1 "" {XPPMATH 20 "+3I\"xG6\",&*&I%t010GF$\"\"\"I%t0
20GF$F(!\"\"I%P022GF$F(!\"#*&,0*(I\"tGF$F(F'F(I)Unknown4GF$F(F**(F0F(F
)F(I%t110GF$F(F(*(F0F(F)F(I%t120GF$F(F(*(F0F(F)F(F1F(F(*&I%P012GF$F(F0
F(F(*&F)F(I%tt10GF$F(F(*&F)F(I%tt20GF$F(F(F(F0F*F*,$*&,<*(F0\"\"#F)F(F
3F(F**(F0FAF)F(F5F(F**(F0FAF3F(F1F(F**(F0FAF5F(F1F(F**&F0FAF1FAF**&I%P
112GF$F(F0FAF(*&I%P122GF$F(F0FAF**(F0F(F:F(F1F(F**(F0F(F<F(F1F(F**&I%P
t12GF$F(F0F(F(F9F*F;F*I%Pt22GF$F*F(F0F,F*\"\"!,$*&,:*(F0\"\"$F)F(F3F(F
**(F0FTF)F(F5F(F**(F0FTF3F(F1F(F**(F0FTF5F(F1F(F**&FGF(F0FTF(*&FIF(F0F
TF,FJF*FKF*FLF(F9F*F;F*FNF,F(F0!\"$F*F(,$*&,:*(F0\"\"%F)F(F3F(F**(F0Fi
nF)F(F5F(F**(F0FinF3F(F1F(F**(F0FinF5F(F1F(F**&FGF(F0FinF(*&FIF(F0FinF
ZFJF*FKF*FLF(F9F*F;F*FNFZF(F0!\"%F*FA,$*&,:*(F0\"\"&F)F(F3F(F**(F0FdoF
)F(F5F(F**(F0FdoF3F(F1F(F**(F0FdoF5F(F1F(F**&FGF(F0FdoF(*&FIF(F0FdoF_o
FJF*FKF*FLF(F9F*F;F*FNF_oF(F0!\"&F*FT,$*&,:*(F0\"\"'F)F(F3F(F**(F0F_pF
)F(F5F(F**(F0F_pF3F(F1F(F**(F0F_pF5F(F1F(F**&FGF(F0F_pF(*&FIF(F0F_pFjo
FJF*FKF*FLF(F9F*F;F*FNFjoF(F0!\"'F*Fin-I\"OG6$%*protectedGI(_syslibGF$
6#F(Fdo" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQ01G6\"\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">I%EQ02G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 
">I%EQ03G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQ04G6\"\"\"!" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQ05G6\"*&,&*(I\"tGF$\"\"#I%t010GF$
\"\"\"I%t020GF$F+!\"\"*&I%P022GF$F+F(F)F+F+F(!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%EQ06G6\"*&,&*(I\"tGF$\"\"#I%t010GF$\"\"\"I%t020GF$F+!
\"\"*&I%P022GF$F+F(F)F+F+F(!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P0
22G6\"*&I%t010GF$\"\"\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "*&,&*&,(*&,(I%t110G6\"\"\"\"I%t120GF)F*I)Unknown3GF)F*F*I
%t010GF)F*F**&I%t020GF)F*F,F*!\"\"I%P012GF)F*F*I\"tGF)F*F**&F-F*,&I%tt
10GF)F*I%tt20GF)F*F*F*F*F2F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*&,(
*&,(I%t110G6\"\"\"\"I%t120GF)F*I)Unknown4GF)F*F*I%t020GF)F*F**&I%t010G
F)F*F,F*!\"\"I%P012GF)F*F*I\"tGF)F*F**&F-F*,&I%tt10GF)F*I%tt20GF)F*F*F
*F*F2F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 11 "Studty at 1" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "series(SpectralCurve(Diag1Sh
eet1),x=1,2);\n" }{MPLTEXT 1 0 42 "series(SpectralCurve(Diag1Sheet2),x
=1,2);\n" }{MPLTEXT 1 0 57 "EQ11:=residue((x-1)^(2)*SpectralCurve(Diag
1Sheet1),x=1);\n" }{MPLTEXT 1 0 57 "EQ12:=residue((x-1)^(2)*SpectralCu
rve(Diag1Sheet2),x=1);\n" }{MPLTEXT 1 0 57 "EQ13:=residue((x-1)^(1)*Sp
ectralCurve(Diag1Sheet1),x=1);\n" }{MPLTEXT 1 0 57 "EQ14:=residue((x-1
)^(1)*SpectralCurve(Diag1Sheet2),x=1);\n" }{MPLTEXT 1 0 57 "EQ15:=resi
due((x-1)^(0)*SpectralCurve(Diag1Sheet1),x=1):\n" }{MPLTEXT 1 0 57 "EQ
16:=residue((x-1)^(0)*SpectralCurve(Diag1Sheet2),x=1):\n" }{MPLTEXT 1 
0 58 "EQ17:=residue((x-1)^(-1)*SpectralCurve(Diag1Sheet1),x=1):\n" }
{MPLTEXT 1 0 58 "EQ18:=residue((x-1)^(-1)*SpectralCurve(Diag1Sheet2),x
=1):\n" }{MPLTEXT 1 0 24 "P122:=solve(EQ13,P122);\n" }{MPLTEXT 1 0 16 
"simplify(EQ13);\n" }{MPLTEXT 1 0 16 "simplify(EQ14);\n" }{MPLTEXT 1 
0 16 "simplify(EQ15);\n" }{MPLTEXT 1 0 15 "simplify(EQ16);" }{MPLTEXT 
1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "++,&I\"xG6\"\"\"\"!\"\"F&,(
*$I%t110GF%\"\"#F&*&,&F*F&I%t120GF%F&F&F*F&F'I%P122GF%F&!\"#,**&F*F&I)
Unknown5GF%F&F+*&F-F&F3F&F'*&,(I%t010GF%F&I%t020GF%F&*&,&I%tt10GF%F&I%
tt20GF%F&F&,&F&F&I\"tGF%F'F'F&F&F*F&F'I%P112GF%F&F',0*$F3F+F&*&F6F&F3F
&F'*&,(F7F'F8F'*(F:F&,&F'F&F>F&F'F=F'F&F&F*F&F'*&F7F&F8F&F&I%P012GF%F&
*&I%Pt22GF%F&FFF0F&*&I%Pt12GF%F&F=F'F&\"\"!-I\"OG6$%*protectedGI(_sysl
ibGF%6#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "++,&I\"xG6\"\"\"\"!\"\"F&
,(*$I%t120GF%\"\"#F&*&,&I%t110GF%F&F*F&F&F*F&F'I%P122GF%F&!\"#,**&F*F&
I)Unknown6GF%F&F+*&F-F&F3F&F'*&,(I%t010GF%F&I%t020GF%F&*&,&I%tt10GF%F&
I%tt20GF%F&F&,&F&F&I\"tGF%F'F'F&F&F*F&F'I%P112GF%F&F',0*$F3F+F&*&F6F&F
3F&F'*&,(F7F'F8F'*(F:F&,&F'F&F>F&F'F=F'F&F&F*F&F'*&F7F&F8F&F&I%P012GF%
F&*&I%Pt22GF%F&FFF0F&*&I%Pt12GF%F&F=F'F&\"\"!-I\"OG6$%*protectedGI(_sy
slibGF%6#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQ11G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQ12G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I%EQ13G6\"*&,.*(I\"tGF$\"\"#I%t110GF$\"\"\"I%t120GF$F+!
\"\"*&I%P122GF$F+F(F)F+*(F(F+F*F+F,F+F)*&F/F+F(F+!\"#*&F*F+F,F+F-F/F+F
+,(*$F(F)F+F(F2F+F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQ14G6\"*&,.
*(I\"tGF$\"\"#I%t110GF$\"\"\"I%t120GF$F+!\"\"*&I%P122GF$F+F(F)F+*(F(F+
F*F+F,F+F)*&F/F+F(F+!\"#*&F*F+F,F+F-F/F+F+,(*$F(F)F+F(F2F+F+F-" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I%P122G6\"*&I%t110GF$\"\"\"I%t120GF$F'"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20
 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*&,.*&,(I%t010G6\"!\"\"I%
t020GF)F*I)Unknown5GF)\"\"\"F-I\"tGF)F-F-F(F-F+F-I%tt10GF)F-I%tt20GF)F
-F,F*F-I%t110GF)F-F-*&,&*&I%t120GF)F-F,F-F*I%P112GF)F-F-,&F*F-F.F-F-F-
F-F7F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&*&,.*&,(I%t010G6\"!\"\"I%t
020GF)F*I)Unknown6GF)\"\"\"F-I\"tGF)F-F-F(F-F+F-I%tt10GF)F-I%tt20GF)F-
F,F*F-I%t120GF)F-F-*&,&*&I%t110GF)F-F,F-F*I%P112GF)F-F-,&F*F-F.F-F-F-F
-F7F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 17 "Study at lambda=t" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "series(SpectralCurve(DiagtSh
eet1),x=t,3);\n" }{MPLTEXT 1 0 42 "series(SpectralCurve(DiagtSheet2),x
=t,3);\n" }{MPLTEXT 1 0 57 "EQt1:=residue((x-t)^(2)*SpectralCurve(Diag
tSheet1),x=t);\n" }{MPLTEXT 1 0 57 "EQt2:=residue((x-t)^(2)*SpectralCu
rve(DiagtSheet2),x=t);\n" }{MPLTEXT 1 0 57 "EQt3:=residue((x-t)^(1)*Sp
ectralCurve(DiagtSheet1),x=t);\n" }{MPLTEXT 1 0 57 "EQt4:=residue((x-t
)^(1)*SpectralCurve(DiagtSheet2),x=t);\n" }{MPLTEXT 1 0 57 "EQt5:=resi
due((x-t)^(0)*SpectralCurve(DiagtSheet1),x=t):\n" }{MPLTEXT 1 0 57 "EQ
t6:=residue((x-t)^(0)*SpectralCurve(DiagtSheet2),x=t):\n" }{MPLTEXT 1 
0 58 "EQt7:=residue((x-t)^(-1)*SpectralCurve(DiagtSheet1),x=t):\n" }
{MPLTEXT 1 0 58 "EQt8:=residue((x-t)^(-1)*SpectralCurve(DiagtSheet2),x
=t):\n" }{MPLTEXT 1 0 32 "Pt22:=factor(solve(EQt3,Pt22));\n" }{MPLTEXT
 1 0 16 "simplify(EQt3);\n" }{MPLTEXT 1 0 16 "simplify(EQt4);\n" }
{MPLTEXT 1 0 16 "simplify(EQt5);\n" }{MPLTEXT 1 0 15 "simplify(EQt6);"
 }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "+-,&I\"xG6\"\"\"
\"I\"tGF%!\"\",(*$I%tt10GF%\"\"#F&*&,&F+F&I%tt20GF%F&F&F+F&F(I%Pt22GF%
F&!\"#,**&F+F&I)Unknown5GF%F&F,*&F.F&F4F&F(*&,&*&,&I%t010GF%F&I%t020GF
%F&F&F'F(F&*&,&I%t110GF%F&I%t120GF%F&F&,&F(F&F'F&F(F&F&F+F&F(I%Pt12GF%
F&F(,0*$F4F,F&*&F7F&F4F&F(*&,&*&F9F&F'F1F(*&F=F&F@F1F(F&F+F&F(*(F:F&F;
F&F'F1F&*&I%P012GF%F&F'F(F&*(F>F&F?F&F@F1F&*&I%P112GF%F&F@F(F&\"\"!,.*
&FFF&F4F&F(*&,&*&F9F&F'!\"$F&*&F=F&F@FUF&F&F+F&F(*(F:F&F;F&F'FUF1*&FKF
&F'F1F(*(F>F&F?F&F@FUF1*&FNF&F@F1F(F&-I\"OG6$%*protectedGI(_syslibGF%6
#F&F," }}{PARA 11 "" 1 "" {XPPMATH 20 "+-,&I\"xG6\"\"\"\"I\"tGF%!\"\",
(*$I%tt20GF%\"\"#F&*&,&I%tt10GF%F&F+F&F&F+F&F(I%Pt22GF%F&!\"#,**&F+F&I
)Unknown6GF%F&F,*&F.F&F4F&F(*&,&*&,&I%t010GF%F&I%t020GF%F&F&F'F(F&*&,&
I%t110GF%F&I%t120GF%F&F&,&F(F&F'F&F(F&F&F+F&F(I%Pt12GF%F&F(,0*$F4F,F&*
&F7F&F4F&F(*&,&*&F9F&F'F1F(*&F=F&F@F1F(F&F+F&F(*(F:F&F;F&F'F1F&*&I%P01
2GF%F&F'F(F&*(F>F&F?F&F@F1F&*&I%P112GF%F&F@F(F&\"\"!,.*&FFF&F4F&F(*&,&
*&F9F&F'!\"$F&*&F=F&F@FUF&F&F+F&F(*(F:F&F;F&F'FUF1*&FKF&F'F1F(*(F>F&F?
F&F@FUF1*&FNF&F@F1F(F&-I\"OG6$%*protectedGI(_syslibGF%6#F&F," }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I%EQt1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I%EQt2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQt3G6\"*&,.*
(I\"tGF$\"\"%I%tt10GF$\"\"\"I%tt20GF$F+!\"\"*&I%Pt22GF$F+F(F)F+*(F(\"
\"$F*F+F,F+\"\"#*&F/F+F(F1!\"#*(F(F2F*F+F,F+F-*&F/F+F(F2F+F+,(*$F(F)F+
*$F(F1F4*$F(F2F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%EQt4G6\"*&,.*(I
\"tGF$\"\"%I%tt10GF$\"\"\"I%tt20GF$F+!\"\"*&I%Pt22GF$F+F(F)F+*(F(\"\"$
F*F+F,F+\"\"#*&F/F+F(F1!\"#*(F(F2F*F+F,F+F-*&F/F+F(F2F+F+,(*$F(F)F+*$F
(F1F4*$F(F2F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Pt22G6\"*&I%tt10GF
$\"\"\"I%tt20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(,(*&,(*&
I%tt10G6\"\"\"\"I)Unknown5GF(F)F)*&I%tt20GF(F)F*F)!\"\"I%Pt12GF(F)F)I
\"tGF(\"\"#F)*&,(*&,,I%t010GF(F-I%t020GF(F-I%t110GF(F-I%t120GF(F-F*F-F
)F'F)F)F+F)F.F-F)F/F)F)*&F'F),&F5F)F6F)F)F)F)F/F-,&F-F)F/F)F-" }}
{PARA 11 "" 1 "" {XPPMATH 20 "*(,(*&,(*&I%tt10G6\"\"\"\"I)Unknown6GF(F
)!\"\"*&I%tt20GF(F)F*F)F)I%Pt12GF(F)F)I\"tGF(\"\"#F)*&,(*&,,I%t010GF(F
+I%t020GF(F+I%t110GF(F+I%t120GF(F+F*F+F)F-F)F)F&F)F.F+F)F/F)F)*&F-F),&
F5F)F6F)F)F)F)F/F+,&F+F)F/F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 
17 "Study at infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "se
ries(SpectralCurve(DiaginftySheet1),x=infinity,3);\n" }{MPLTEXT 1 0 
53 "series(SpectralCurve(DiaginftySheet2),x=infinity,3);\n" }{MPLTEXT 
1 0 78 "EQinftDiag1Sheet1:=residue(x^(-2)*SpectralCurve(DiaginftySheet
1),x=infinity);\n" }{MPLTEXT 1 0 69 "EQinfty2:=residue(x^(-2)*Spectral
Curve(DiaginftySheet2),x=infinity);\n" }{MPLTEXT 1 0 69 "EQinfty3:=res
idue(x^(-1)*SpectralCurve(DiaginftySheet1),x=infinity);\n" }{MPLTEXT 
1 0 69 "EQinfty4:=residue(x^(-1)*SpectralCurve(DiaginftySheet2),x=infi
nity);\n" }{MPLTEXT 1 0 68 "EQinfty5:=residue(x^(0)*SpectralCurve(Diag
inftySheet1),x=infinity);\n" }{MPLTEXT 1 0 68 "EQinfty6:=residue(x^(0)
*SpectralCurve(DiaginftySheet2),x=infinity);\n" }{MPLTEXT 1 0 68 "EQin
fty7:=residue(x^(1)*SpectralCurve(DiaginftySheet1),x=infinity);\n" }
{MPLTEXT 1 0 68 "EQinfty8:=residue(x^(1)*SpectralCurve(DiaginftySheet2
),x=infinity);\n" }{MPLTEXT 1 0 68 "EQinfty9:=residue(x^(2)*SpectralCu
rve(DiaginftySheet1),x=infinity);\n" }{MPLTEXT 1 0 78 "EQinftDiag1Shee
t10:=residue(x^(2)*SpectralCurve(DiaginftySheet2),x=infinity);\n" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&,(I%P012G6\"\"
\"\"I%P112GF&F'I%Pt12GF&F'F'I\"xGF&!\"\"F'*&,0*$I)tinfty10GF&\"\"#F'*&
,.I%t010GF&F'I%t020GF&F'I%t110GF&F'I%t120GF&F'I%tt10GF&F'I%tt20GF&F'F'
F/F'F'*&F3F'F4F'F'*&F5F'F6F'F'F(F'*&F7F'F8F'F'*&F)F'I\"tGF&F'F'F'F*!\"
#F'-I\"OG6$%*protectedGI(_syslibGF&6#*$F*!\"$F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",(*&,(I%P012G6\"\"\"\"I%P112GF&F'I%Pt12GF&F'F'I\"xGF&!\"
\"F'*&,0*$I)tinfty20GF&\"\"#F'*&,.I%t010GF&F'I%t020GF&F'I%t110GF&F'I%t
120GF&F'I%tt10GF&F'I%tt20GF&F'F'F/F'F'*&F3F'F4F'F'*&F5F'F6F'F'F(F'*&F7
F'F8F'F'*&F)F'I\"tGF&F'F'F'F*!\"#F'-I\"OG6$%*protectedGI(_syslibGF&6#*
$F*!\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2EQinftDiag1Sheet1G6\"\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty2G6\"\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">I)EQinfty3G6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)EQinfty4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)
EQinfty5G6\",(I%P012GF$!\"\"I%P112GF$F'I%Pt12GF$F'" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I)EQinfty6G6\",(I%P012GF$!\"\"I%P112GF$F'I%Pt12GF$F'" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty7G6\",0*$I)tinfty10GF$\"\"#!
\"\"*&,.I%t010GF$\"\"\"I%t020GF$F-I%t110GF$F-I%t120GF$F-I%tt10GF$F-I%t
t20GF$F-F-F'F-F)*&F,F-F.F-F)*&F/F-F0F-F)I%P112GF$F)*&F1F-F2F-F)*&I%Pt1
2GF$F-I\"tGF$F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)EQinfty8G6\",0*$
I)tinfty20GF$\"\"#!\"\"*&,.I%t010GF$\"\"\"I%t020GF$F-I%t110GF$F-I%t120
GF$F-I%tt10GF$F-I%tt20GF$F-F-F'F-F)*&F,F-F.F-F)*&F/F-F0F-F)I%P112GF$F)
*&F1F-F2F-F)*&I%Pt12GF$F-I\"tGF$F-F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I)EQinfty9G6\",0*&I)tinfty10GF$\"\"\"I(UnknownGF$F(\"\"#*&,.I%t010GF$
F(I%t020GF$F(I%t110GF$F(I%t120GF$F(I%tt10GF$F(I%tt20GF$F(F(F)F(F(*&,(F
/F(F0F(*&,&F1!\"\"F2F7F(I\"tGF$F(F7F(F'F(F7*&F/F(F0F(!\"#I%P112GF$F7*(
F8F(F1F(F2F(F:*&I%Pt12GF$F(F8F*F7" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3
EQinftDiag1Sheet10G6\",0*&I)tinfty20GF$\"\"\"I)Unknown2GF$F(\"\"#*&,.I
%t010GF$F(I%t020GF$F(I%t110GF$F(I%t120GF$F(I%tt10GF$F(I%tt20GF$F(F(F)F
(F(*&,(F/F(F0F(*&,&F1!\"\"F2F7F(I\"tGF$F(F7F(F'F(F7*&F/F(F0F(!\"#I%P11
2GF$F7*(F8F(F1F(F2F(F:*&I%Pt12GF$F(F8F*F7" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 41 "CoherenceEquation2:=simplify(-EQinfty5);\n" }
{MPLTEXT 1 0 40 "CoherenceEquation1:=CoherenceEquation1;\n" }{MPLTEXT 
1 0 75 "factor(simplify(EQinfty7-EQinfty8+(tinfty10-tinfty20)*Coherenc
eEquation1));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation2G6
\",(I%P012GF$\"\"\"I%P112GF$F'I%Pt12GF$F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I3CoherenceEquation1G6\",2I)tinfty10GF$\"\"\"I)tinfty20G
F$F'I%t010GF$F'I%t020GF$F'I%t110GF$F'I%t120GF$F'I%tt10GF$F'I%tt20GF$F'
" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "CoherenceEquation3:=Pt12*t+t010*t020+t110*t120+tt10*t
t20-tinfty10*tinfty20+P112;\n" }{MPLTEXT 1 0 38 "CoherenceEquation2 :=
 P012+P112+Pt12;\n" }{MPLTEXT 1 0 71 "CoherenceEquation1 := tinfty10+t
infty20+t010+t020+t110+t120+tt10+tt20;\n" }{MPLTEXT 1 0 59 "solve(\{Co
herenceEquation1,CoherenceEquation2\},\{P012,P112\});" }{MPLTEXT 1 0 
1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation3G6\",.*&I%
Pt12GF$\"\"\"I\"tGF$F(F(*&I%t010GF$F(I%t020GF$F(F(*&I%t110GF$F(I%t120G
F$F(F(*&I%tt10GF$F(I%tt20GF$F(F(*&I)tinfty10GF$F(I)tinfty20GF$F(!\"\"I
%P112GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation2G6\",
(I%P012GF$\"\"\"I%P112GF$F'I%Pt12GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I3CoherenceEquation1G6\",2I)tinfty10GF$\"\"\"I)tinfty20GF$F'I%t010G
F$F'I%t020GF$F'I%t110GF$F'I%t120GF$F'I%tt10GF$F'I%tt20GF$F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 215 58 "Summary of coefficients and addit
ional coherence relations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "P022:=P022;\n" }{MPLTEXT 1 0 12 "P122:=P122;\n" }{MPLTEXT 1 0 12 "
Pt22:=Pt22;\n" }{MPLTEXT 1 0 12 "P011:=P011;\n" }{MPLTEXT 1 0 12 "P111
:=P111;\n" }{MPLTEXT 1 0 12 "Pt11:=Pt11;\n" }{MPLTEXT 1 0 80 "Coherenc
eEquation1 := tinftDiag1Sheet10+tt10+tt20+t010+t020+tinfty20+t110+t120
;\n" }{MPLTEXT 1 0 38 "CoherenceEquation2 := Pt12+P012+P112;\n" }
{MPLTEXT 1 0 93 "CoherenceEquation3 := -tinftDiag1Sheet10*tinfty20+tt2
0*tt10+Pt12*t+t010*t020+P112+t120*t110;\n" }{MPLTEXT 1 0 76 "Coherence
Equation3-(P112+t*Pt12+P022+P122+Pt22-tinftDiag1Sheet10*tinfty20);\n" 
}{MPLTEXT 1 0 66 "CoherenceEquation1- (P011+P111+Pt11+ tinftDiag1Sheet
10+tinfty20);\n" }{MPLTEXT 1 0 13 "P2:=unapply( " }{MPLTEXT 1 0 20 "P2
(lambda),lambda);\n" }{MPLTEXT 1 0 32 "P1:=unapply( P1(lambda),lambda)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P022G6\"*&I%t010GF$\"\"\"I%t020G
F$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P122G6\"*&I%t110GF$\"\"\"I%t1
20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Pt22G6\"*&I%tt10GF$\"\"\"I
%tt20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P011G6\",&I%t010GF$\"\"
\"I%t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P111G6\",&I%t110GF$
\"\"\"I%t120GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Pt11G6\",&I%tt10
GF$\"\"\"I%tt20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEqua
tion1G6\",2I2tinftDiag1Sheet10GF$\"\"\"I%tt10GF$F'I%tt20GF$F'I%t010GF$
F'I%t020GF$F'I)tinfty20GF$F'I%t110GF$F'I%t120GF$F'" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I3CoherenceEquation2G6\",(I%P012GF$\"\"\"I%P112GF$F'I%P
t12GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3CoherenceEquation3G6\",.*
&I%Pt12GF$\"\"\"I\"tGF$F(F(*&I%t010GF$F(I%t020GF$F(F(*&I%t110GF$F(I%t1
20GF$F(F(*&I%tt10GF$F(I%tt20GF$F(F(*&I)tinfty20GF$F(I2tinftDiag1Sheet1
0GF$F(!\"\"I%P112GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">
I#P2G6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,.*(I%t010GF$\"\"
\"I%t020GF$F.9$!\"#F.*&I%P012GF$F.F0!\"\"F.*(I%t110GF$F.I%t120GF$F.,&F
0F.F4F.F1F.*&I%P112GF$F.F8F4F.*(I%tt10GF$F.I%tt20GF$F.,&F0F.I\"tGF$F4F
1F.*&I%Pt12GF$F.F>F4F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6
\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,(*&,&I%t010GF$\"\"\"I%t
020GF$F/F/9$!\"\"F/*&,&I%t110GF$F/I%t120GF$F/F/,&F1F/F2F/F2F/*&,&I%tt1
0GF$F/I%tt20GF$F/F/,&F1F/I\"tGF$F2F2F/F$F$F$" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 216 136 "We have 3 undetermined coefficients: P012, P112 and P
t12 but they are related by 2 coherence equations: sum of monodromies \+
is null and \n" }{TEXT 216 46 "P112+t*Pt12+P022+P122+Pt22-tinfty10*tin
fty20=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2
 33 1 1 }