# Need to read "Homomorphisms.txt", change of variables and cyclic vector are in that file.

read "Homomorphisms.txt";


# C(x,y)[Dx,Dy]-module of G3 with basis [G3(a,b,x,y), DxG3, DyG3]

M := [[x, y], [Dx, Dy], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = -(-a-4*a*y+a*x*y+12*a*x*y^2-a^2-4*a^2*y+a^2*x*y+12*a^2*x*y^2+3*b*x*y^2+4*y^2*b+y*b+2*a*b*y+8*a*b*y^2+2*a*b*x*y+12*a*b*y^2*x+3*y^2*b^2*x+4*y^2*b^2+y*b^2)/(x*(27*x^2*y^2-18*x*y-4*x-4*y-1)), (1, 3) = -(6*a*x*y+2*a*x+6*a^2*x*y+2*a^2*x+6*y*b*x+2*y*b+15*a*b*x*y+4*a*b*x+a*b+4*a*b*y+6*y*b^2*x+2*y*b^2)/(27*x^2*y^2-18*x*y-4*x-4*y-1), (2, 1) = 1, (2, 2) = -(b-4*a*x+4*y*b+36*a*x^2*y^2+2*a*x^2*y-8*a*x*y^2-18*a*x*y+4*b*x^2*y-4*b*x*y^2+18*b*x^2*y^2+y*b*x-27*x*y+2*x^2*y-4*x*y^2+48*x^2*y^2-6*x-4*y-1)/(x*(27*x^2*y^2-18*x*y-4*x-4*y-1)), (2, 3) = -(-2*y+4*x-a+6*x*y-2*y*b+8*b*x-4*a*y+4*a*x+18*y*b*x+9*a*x*y)*x/(27*x^2*y^2-18*x*y-4*x-4*y-1), (3, 1) = 0, (3, 2) = y*(-1-6*y-a-8*y^2+x*y-2*y*b-8*a*y+6*x*y^2-8*y^2*b+2*y*b*x-16*a*y^2+a*x*y)/((27*x^2*y^2-18*x*y-4*x-4*y-1)*x), (3, 3) = -(4*y-2*x-b+6*x*y+4*y*b-4*b*x+8*a*y-2*a*x+9*y*b*x+18*a*x*y)*y/(27*x^2*y^2-18*x*y-4*x-4*y-1)}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = -(6*a*x*y+2*a*x+6*a^2*x*y+2*a^2*x+6*y*b*x+2*y*b+15*a*b*x*y+4*a*b*x+a*b+4*a*b*y+6*y*b^2*x+2*y*b^2)/(27*x^2*y^2-18*x*y-4*x-4*y-1), (1, 3) = -(3*a*x^2*y+4*a*x^2+a*x+3*a^2*x^2*y+4*a^2*x^2+a^2*x+12*b*x^2*y+y*b*x-4*b*x-b+2*a*b*x+2*a*b*x*y+8*a*b*x^2+12*a*b*x^2*y+12*b^2*x^2*y+y*b^2*x-4*b^2*x-b^2)/((27*x^2*y^2-18*x*y-4*x-4*y-1)*y), (2, 1) = 0, (2, 2) = -(-2*y+4*x-a+6*x*y-2*y*b+8*b*x-4*a*y+4*a*x+18*y*b*x+9*a*x*y)*x/(27*x^2*y^2-18*x*y-4*x-4*y-1), (2, 3) = -(1+6*x+b-x*y+8*x^2+8*b*x+2*a*x-6*x^2*y-y*b*x+16*b*x^2-2*a*x*y+8*a*x^2)*x/((27*x^2*y^2-18*x*y-4*x-4*y-1)*y), (3, 1) = 1, (3, 2) = -(4*y-2*x-b+6*x*y+4*y*b-4*b*x+8*a*y-2*a*x+9*y*b*x+18*a*x*y)*y/(27*x^2*y^2-18*x*y-4*x-4*y-1), (3, 3) = -(-1+a*x*y+18*a*x^2*y^2+4*a*x*y^2-4*a*x^2*y-18*y*b*x+36*b*x^2*y^2+2*b*x*y^2-8*b*x^2*y-4*x-6*y-27*x*y-4*x^2*y+2*x*y^2+48*x^2*y^2+4*a*x-4*y*b+a)/(y*(27*x^2*y^2-18*x*y-4*x-4*y-1))})]]: