# Sasaki Yoshida system for X_8

l := -(-x^2*y^2 - y^2 + 2)/(x*y*(-x^2 + 1));
m := -(-x^2*y^2 - x^2 + 2)/(x*y*(-y^2 + 1));
a := ((-2*y^4 + 4*y^2)*x^6 + (-y^6 + 8*y^4 - 10*y^2 - 3)*x^4 + (2*y^6 - 8*y^4 + 4*y^2 + 8)*x^2 - y^4 + 5*y^2 - 6)/((y^2 - 1)*x*(x^2 + y^2 - 2)*(x^2*y^2 - 1)*(x^2 - 1));
b := ((3*x^2 + 1)*y^6 + (-x^6 + 3*x^4 - 7*x^2 - 7)*y^4 + (-x^2 + 13)*y^2 + 2*x^2 - 6)/((x^2 + y^2 - 2)*(x^2*y^2 - 1)*(x^2 - 1)^2*y);
c := ((3*y^2 + 1)*x^6 + (-y^6 + 3*y^4 - 7*y^2 - 7)*x^4 + (-y^2 + 13)*x^2 + 2*y^2 - 6)/((x^2 + y^2 - 2)*(x^2*y^2 - 1)*(y^2 - 1)^2*x);
d := ((-2*x^4 + 4*x^2)*y^6 + (-x^6 + 8*x^4 - 10*x^2 - 3)*y^4 + (2*x^6 - 8*x^4 + 4*x^2 + 8)*y^2 - x^4 + 5*x^2 - 6)/((y^2 - 1)*y*(x^2 + y^2 - 2)*(x^2*y^2 - 1)*(x^2 - 1));
p := -2*(x^2 - y^2)/((-x^2 + 1)^2*(-y^2 + 1));
q := -2*(-x^2 + y^2)/((-x^2 + 1)*(-y^2 + 1)^2);

L0 := diff(F(x, y), x, x) - l*diff(F(x, y), x, y) - a*diff(F(x, y), x) - b*diff(F(x, y), y) - p*F(x, y);
R0 := diff(F(x, y), y, y) - m*diff(F(x, y), x, y) - c*diff(F(x, y), x) - d*diff(F(x, y), y) - q*F(x, y);

DE := x*(x - 1)^4*(x + 1)^4*(x^2*y^2 - 2*y^2 + 1)*(x^2 + y^2 - 2)^3*(x*y - 1)^3*(x*y + 1)^3*diff(F(x), x, x, x, x)
+ (x - 1)^3*(x + 1)^3*(12*x^8*y^4 + 3*x^6*y^6 - 49*x^6*y^4 - 16*x^4*y^6 + 11*x^6*y^2 + 77*x^4*y^4 + 16*x^2*y^6 - 23*x^4*y^2 - 51*x^2*y^4 - 5*x^4 + 26*x^2*y^2 + 2*y^4 - 5*y^2 + 2)*(x^2 + y^2 - 2)^2*(x*y - 1)^2*(x*y + 1)^2*diff(F(x), x, x, x)
+ x*(x - 1)^2*(x + 1)^2*(x^2 + y^2 - 2)*(x*y - 1)*(x*y + 1)*(37*x^12*y^6 + 29*x^10*y^8 + x^8*y^10 - 242*x^10*y^6 - 146*x^8*y^8 + 6*x^6*y^10 + 25*x^10*y^4 + 608*x^8*y^6 + 218*x^6*y^8 + x^4*y^10 - 30*x^8*y^4 - 652*x^6*y^6 - 188*x^4*y^8 - 26*x^2*y^10 - 48*x^8*y^2 - 138*x^6*y^4 + 387*x^4*y^6 + 169*x^2*y^8 + 162*x^6*y^2 + 172*x^4*y^4 - 274*x^2*y^6 + 8*y^8 + 4*x^6 - 141*x^4*y^2 + 73*x^2*y^4 - 44*y^6 - 16*x^4 - 14*x^2*y^2 + 78*y^4 + 20*x^2 - 49*y^2 + 10)*diff(F(x), x, x)
+ (x - 1)*(x + 1)*(29*x^18*y^8 + 46*x^16*y^10 + 14*x^14*y^12 - 274*x^16*y^8 - 368*x^14*y^10 - 152*x^12*y^12 - 37*x^10*y^14 + 11*x^16*y^6 + 1079*x^14*y^8 + 1178*x^12*y^10 + 365*x^10*y^12 - 6*x^8*y^14 + 57*x^14*y^6 - 1932*x^12*y^8 - 1282*x^10*y^10 + 191*x^8*y^12 + 113*x^6*y^14 - 78*x^14*y^4 - 903*x^12*y^6 + 473*x^10*y^8 - 963*x^8*y^10 - 717*x^6*y^12 + 2*x^4*y^14 + 489*x^12*y^4 + 3192*x^10*y^6 + 3024*x^8*y^8 + 1578*x^6*y^10 - 205*x^4*y^12 + 32*x^12*y^2 - 1069*x^10*y^4 - 4329*x^8*y^6 - 1563*x^6*y^8 + 1239*x^4*y^10 - 200*x^10*y^2 + 550*x^8*y^4 + 200*x^6*y^6 - 2796*x^4*y^8 + 92*x^2*y^10 + 551*x^8*y^2 + 1571*x^6*y^4 + 2879*x^4*y^6 - 620*x^2*y^8 - 8*y^10 - 12*x^8 - 816*x^6*y^2 - 1310*x^4*y^4 + 1579*x^2*y^6 + 60*y^8 + 26*x^6 + 33*x^4*y^2 - 1870*x^2*y^4 - 166*y^6 + 94*x^4 + 1012*x^2*y^2 + 205*y^4 - 200*x^2 - 108*y^2 + 20)*diff(F(x), x)
+ x^3*(3*x^16*y^8 + 4*x^14*y^10 + x^12*y^12 - 33*x^14*y^8 - 21*x^12*y^10 + 57*x^10*y^12 + 45*x^8*y^14 + 5*x^14*y^6 + 124*x^12*y^8 + 8*x^10*y^10 - 66*x^8*y^12 + 65*x^6*y^14 - 6*x^12*y^6 - 355*x^10*y^8 - 739*x^8*y^10 - 917*x^6*y^12 - 207*x^4*y^14 - 14*x^12*y^4 + 57*x^10*y^6 + 1628*x^8*y^8 + 3032*x^6*y^10 + 1138*x^4*y^12 - 49*x^2*y^14 + 61*x^10*y^4 - 445*x^8*y^6 - 2937*x^6*y^8 - 1313*x^4*y^10 + 760*x^2*y^12 + 2*y^14 + 4*x^10*y^2 - 193*x^8*y^4 - 432*x^6*y^6 - 2318*x^4*y^8 - 3480*x^2*y^10 + 35*y^12 - 20*x^8*y^2 + 1055*x^6*y^4 + 5491*x^4*y^6 + 6693*x^2*y^8 - 515*y^10 - 46*x^6*y^2 - 2815*x^4*y^4 - 5190*x^2*y^6 + 2235*y^8 + 12*x^6 + 92*x^4*y^2 + 304*x^2*y^4 - 4520*y^6 + 16*x^4 + 1222*x^2*y^2 + 4626*y^4 - 284*x^2 - 2260*y^2 + 400)*F(x);


# Because this system, and the one at the beginning of Section 2, are both related to X8,
# we expect the two systems to be related via some transformation.

# We did not compute such a transformation, but instead ran a test, namely we tested
# if the above DE is a symmetric product (like DE_z and DE_y in Sections 2.1 and 2.2).

# Indeed it is, DE is the symmetric_product of:

_Envdiffopdomain := [Dx,x]:

L_Sasaki_Yoshida :=
Dx^2+(-1/2/(x+1)-1/2/(x-1)+y/(x*y+1)+2*x/(x^2+y^2-2)-y^2*x/(x^2*y^2+y^2-2)+y/(x*y-1))*Dx+1/4*(40+8*x^2-16*x^4-2*y*x*(x-1)*(x+1)*(y^2-2)*(x^2+y^2-2)*(x*y-1)*(x*y+1)*(x^2*y^2-2*y
^2+1)*(-2*(x^2+y^2-2)*(y-1)*(y+1)*(x-1)*(x+1)*(x*y-1)*(x*y+1))^(1/2)-124*y^2+142*y^4-77*y^6+20*y^8-2*y^10-80*x^2*y^2-38*x^8*y^10-69*x^4*y^10-120*x^8*y^4-160*x^6*y^6-26*x^6*y^4-\
316*x^4*y^6+106*x^6*y^2+236*x^4*y^4-234*x^2*y^6-80*x^4*y^2+226*x^2*y^4-10*x^12*y^6-52*x^10*y^8+38*x^10*y^6+8*x^8*y^8-78*x^6*y^10+22*x^10*y^4+119*x^8*y^6+181*x^6*y^8+216*x^4*y^8
-20*x^2*y^10-14*x^8*y^2+106*x^2*y^8+x^14*y^8+3*x^12*y^10+x^10*y^12+12*x^10*y^10+10*x^8*y^12+12*x^6*y^12+8*x^4*y^12+x^2*y^12)/(x^2*y^2+y^2-2)^2/(x^2+y^2-2)^2/(x*y+1)^2/(x*y-1)^2
/(x+1)^2/(x-1)^2 ;

# and its conjugate.

# Note: The square-root here has genus 2 but that does not mean the systems are unrelated; DE is defined over C(x^2)
# so we can do a change of variables x -> sqrt(X) and that reduces the genus to 1, like in Section 2.1.