# filename = Z3Mmatrix
# The Pfaffian form for the system $Z_3$ is given as the matrix M.

M[1, 1]:= 0:
M[1, 2]:= dt1:
M[1, 3]:= dt2:
M[1, 4]:= dt3:
M[1, 5]:= 0:
M[1, 6]:= 0:
M[1, 7]:= 0:
M[1, 8]:= 0:

M[2, 1]:= -(1/2)*(-a2-a3+a1)*dt1/((t1-1)*(t1+1)):
M[2, 2]:= a0*t1*dt1/((t1-1)*(t1+1)):
M[2, 3]:= 0:
M[2, 4]:= 0:
M[2, 5]:= -(-t3+t1*t2)*dt1/((t1-1)*(t1+1))+dt2:
M[2, 6]:= -(-t2+t3*t1)*dt1/((t1-1)*(t1+1))+dt3:
M[2, 7]:= -(-t2*t3+t1)*dt1/((t1-1)*(t1+1)):
M[2, 8]:= 0:

M[3, 1]:= (1/2)*(-a2+a3+a1)*dt2/((t2-1)*(t2+1)):
M[3, 2]:= 0:
M[3, 3]:= a0*t2*dt2/((t2-1)*(t2+1)):
M[3, 4]:= 0:
M[3, 5]:= dt1-(-t3+t1*t2)*dt2/((t2-1)*(t2+1)):
M[3, 6]:= (-t2+t3*t1)*dt2/((t2-1)*(t2+1)):
M[3, 7]:= (-t2*t3+t1)*dt2/((t2-1)*(t2+1))+dt3:
M[3, 8]:= 0:
M[4, 1]:= (1/2)*(a2-a3+a1)*dt3/((t3-1)*(t3+1)):
M[4, 2]:= 0:
M[4, 3]:= 0:
M[4, 4]:= a0*t3*dt3/((t3-1)*(t3+1)):
M[4, 5]:= (-t3+t1*t2)*dt3/((t3-1)*(t3+1)):
M[4, 6]:= dt1-(-t2+t3*t1)*dt3/((t3-1)*(t3+1)):
M[4, 7]:= dt2+(-t2*t3+t1)*dt3/((t3-1)*(t3+1)):
M[4, 8]:= 0:

M[5, 1]:= 0:
M[5, 2]:= (1/2)*(-2*a1*t1*t2-t3*a3*t1^2+t3*a3+a2*t1^2*t3+a1*t1^2*t3+a1*t3-a2*t3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+(1/2)*(-2*a1*t1*t2*t3-a2*t1^2+a1*t1^2+a3*t1^2+a2-a3-a1+2*a1*t2^2)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)):

M[5, 3]:= -(1/2)*(-2*a2*t1^2+2*a2*t1*t2*t3-a2*t2^2-a3*t2^2+a1*t2^2+a3-a1+a2)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+(1/2)*(t3*a3+t2^2*t3*a2+a2*t3-t2^2*t3*a3-a1*t3+a1*t2^2*t3-2*a2*t1*t2)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)):

M[5, 4]:= (1/2)*(2*a3*t2*t3-a3*t1*t3^2+t1*a1*t3^2-t1*a1-a3*t1+a2*t1-a2*t1*t3^2)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(1/2)*(t2*a3*t3^2-t2*a2*t3^2+t2*a1*t3^2-2*a3*t1*t3-t2*a1+t2*a2+t2*a3)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)):

M[5, 5]:= (a0*t1^3+t2*t3*t1^2-a0*t2*t3*t1^2-t1*t3^2-a0*t1-t1*t2^2+t2*a0*t3+t2*t3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(t2*t3^2-t3*t1*t2^2+t1*a0*t3*t2^2-t3*a0*t1-t3*t1-a0*t2^3+t2*t1^2+a0*t2)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)):

M[5, 6]:= dt1/((t1-1)*(t1+1))+(-t2*t3+t1)*(-t2+t3*t1)*(a0+1)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)):

M[5, 7]:= (-t2*t3+t1)*(-t2+t3*t1)*(a0+1)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+dt2/((t2-1)*(t2+1)):

M[5, 8]:= -(-t2+t3*t1)*dt1/((t1-1)*(t1+1))+(-t2*t3+t1)*dt2/((t2-1)*(t2+1))+dt3:


M[6, 1]:= 0:
M[6, 2]:= (1/2)*(-2*a1*t1*t3+t2*a2+t2*a3*t1^2-t2*a2*t1^2+t2*a1*t1^2-t2*a3+t2*a1)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+(1/2)*(a2*t1^2-a3*t1^2+a1*t1^2-2*a1*t1*t2*t3-a1+2*t3^2*a1+a3-a2)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[6, 3]:= (1/2)*(2*a2*t2*t3+t1*a1*t2^2-a2*t1*t2^2+a3*t1-a2*t1-t1*a1-a3*t1*t2^2)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(1/2)*(t3*a3+t2^2*t3*a2+a2*t3-t2^2*t3*a3-a1*t3+a1*t2^2*t3-2*a2*t1*t2)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[6, 4]:= -(1/2)*(-2*a3*t1^2+2*t3*t1*t2*a3-a2*t3^2+t3^2*a1+a2-a1-t3^2*a3+a3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+(1/2)*(t2*a3*t3^2-t2*a2*t3^2+t2*a1*t3^2-2*a3*t1*t3-t2*a1+t2*a2+t2*a3)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[6, 5]:= dt1/((t1-1)*(t1+1))+(-t2*t3+t1)*(-t3+t1*t2)*(a0+1)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[6, 6]:= (a0*t1^3+t2*t3*t1^2-a0*t2*t3*t1^2-t1*t3^2-a0*t1-t1*t2^2+t2*a0*t3+t2*t3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(t3*t1^2-t1*t2*t3^2-t1*t2-t2*a0*t1+t3^2*a0*t2*t1+t2^2*t3+a0*t3-a0*t3^3)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[6, 7]:= (-t2*t3+t1)*(-t3+t1*t2)*(a0+1)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+dt3/((t3-1)*(t3+1)):

M[6, 8]:= -(-t3+t1*t2)*dt1/((t1-1)*(t1+1))+dt2+(-t2*t3+t1)*dt3/((t3-1)*(t3+1)):


M[7, 1]:= 0:

M[7, 2]:= -(1/2)*(-2*a1*t1*t3+t2*a2+t2*a3*t1^2-t2*a2*t1^2+t2*a1*t1^2-t2*a3+t2*a1)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(1/2)*(-2*a1*t1*t2-t3*a3*t1^2+t3*a3+a2*t1^2*t3+a1*t1^2*t3+a1*t3-a2*t3)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[7, 3]:= -(1/2)*(2*a2*t2*t3+t1*a1*t2^2-a2*t1*t2^2+a3*t1-a2*t1-t1*a1-a3*t1*t2^2)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+(1/2)*(-2*a2*t1*t2*t3-a3*t2^2+a2*t2^2+a1*t2^2+2*a2*t3^2-a2-a1+a3)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[7, 4]:= (1/2)*(-a2*t3^2+t3^2*a3+t3^2*a1-2*t3*t1*t2*a3+a2+2*a3*t2^2-a1-a3)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(1/2)*(2*a3*t2*t3-a3*t1*t3^2+t1*a1*t3^2-t1*a1-a3*t1+a2*t1-a2*t1*t3^2)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[7, 5]:= dt2/((t2-1)*(t2+1))-(-t3+t1*t2)*(-t2+t3*t1)*(a0+1)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[7, 6]:= -(-t3+t1*t2)*(-t2+t3*t1)*(a0+1)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+dt3/((t3-1)*(t3+1)):

M[7, 7]:= -(t2*t3^2-t3*t1*t2^2+t1*a0*t3*t2^2-t3*a0*t1-t3*t1-a0*t2^3+t2*t1^2+a0*t2)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(t3*t1^2-t1*t2*t3^2-t1*t2-t2*a0*t1+t3^2*a0*t2*t1+t2^2*t3+a0*t3-a0*t3^3)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[7, 8]:= dt1-(-t3+t1*t2)*dt2/((t2-1)*(t2+1))-(-t2+t3*t1)*dt3/((t3-1)*(t3+1)):


M[8, 1]:= (1/4)*(2*a1^2*t1*t3^2-a3^2*t2*t3-a2^2*t2*t3-a1^2*t2*t3-2*a2*a3*t1-a3^2*t1^3+a1^2*t1^3+a2^2*t1+a3^2*t1+2*a2*a3*t2*t3+2*a1*a3*t2*t3+2*a1*a2*t2*t3-a2^2*t1^3+2*a2*a3*t1^3+2*a1^2*t1*t2^2+a3^2*t1^2*t2*t3+a2^2*t1^2*t2*t3-3*a1^2*t1^2*t2*t3-2*a2*a3*t1^2*t2*t3+2*a1*a3*t1^2*t2*t3+2*a1*a2*t1^2*t2*t3-a1^2*t1-2*t1*a1*a3*t2^2-2*t1*a1*a2*t2^2-2*a1*a3*t1*t3^2-2*a1*a2*t1*t3^2)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)+(1/4)*(-2*a1*a3*t2-2*a2*a3*t1^2*t2+2*a2*a3*t1*t3+a1^2*t2+a3^2*t2-a2^2*t2+2*a1*a3*t1*t3+2*a1*a2*t1*t3+a2^2*t2^3-a3^2*t2^3-a1^2*t2^3+2*a2^2*t2*t3^2+2*a2^2*t1^2*t2+2*a1*a3*t2^3-a2^2*t1*t3+a3^2*t1*t2^2*t3-2*a1*a2*t1^2*t2-a1^2*t1*t3-a3^2*t1*t3-2*a2*a3*t2*t3^2-2*a1*a2*t2*t3^2+a1^2*t1*t2^2*t3-3*a2^2*t1*t2^2*t3-2*a1*a3*t1*t2^2*t3+2*a1*a2*t1*t2^2*t3+2*a2*a3*t1*t2^2*t3)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)+(1/4)*(2*a1*a2*t3^3-2*a1*a2*t3+2*a3^2*t1^2*t3+2*a3^2*t2^2*t3-2*a1*a3*t1^2*t3-2*a2*a3*t1^2*t3-2*a2*a3*t2^2*t3-3*a3^2*t1*t2*t3^2+a2^2*t1*t2*t3^2+a1^2*t1*t2*t3^2+2*t1*a2*a3*t2+2*t1*a1*a3*t2+2*t1*a1*a2*t2+a3^2*t3^3-a3^2*t3+a2^2*t3+a1^2*t3-a1^2*t3^3-a2^2*t3^3-2*a1*a2*t1*t2*t3^2+2*t1*a1*a3*t2*t3^2+2*t1*a2*a3*t2*t3^2-t1*a3^2*t2-t1*a1^2*t2-t1*a2^2*t2-2*a1*a3*t2^2*t3)*dt3/((t3-1)*(t3+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2):

M[8, 2]:= -(1/2)*(-2*a1-a2*t2^2+a2*t3^2+a1*t2^2+t3^2*a1-t3^2*a3+a3*t2^2)*dt1/(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2-(1/2)*(-t2^2*a0*t3*a1-2*a0*t3*a1*t1^2-t2^2*a0*t3*a2+t2^2*a0*t3*a3+2*t2*t3^2*t1*a0*a1-a2*t1^2*t3-t2^2*t3*a2+t2^2*t3*a3+a1*t2^2*t3+t3*a3*t1^2+a1*t1^2*t3-t3*a3+a2*t3-a1*t3+t2^2*t3*a2*t1^2-a1*t1^2*t2^2*t3-t2^2*t3*a3*t1^2-a1*a0*t1^2*t2^2*t3+t3*t1^2*a0*a2*t2^2-t3*t1^2*a0*a3*t2^2+a3*a0*t1^3*t2-a2*a0*t1^3*t2+a1*a0*t1^3*t2+t1*a0*t2*a2-t1*a0*t2*a3+t1*a0*t2*a1)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)-(1/2)*(-2*a0*t2*a1*t1^2-a0*a3*t1^3*t3+2*t3*t1*t2^2*a0*a1+a3*a0*t1^2*t2*t3^2-a1*a0*t1^2*t2*t3^2-a2*a0*t1^2*t2*t3^2+a2*a0*t1^3*t3+a1*a0*t1^3*t3+t2*a2*t3^2+t2*a1*t3^2-a1*t1^2*t2*t3^2-a2*t1^2*t2*t3^2-t2*a3*t1^2+t2*a2*t1^2+t2*a1*t1^2+a3*t1^2*t2*t3^2-t2*a2+t2*a3-t2*a1-t2*a3*t3^2-t3^2*a0*t2*a3-t3^2*a0*t2*a1+t3^2*a0*t2*a2+a1*a0*t1*t3+a0*a3*t1*t3-a2*a0*t1*t3)*dt3/((t3-1)*(t3+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2):

M[8, 3]:= -(1/2)*(-a0*t3*a2*t1^2+a0*t3*a3*t1^2-a0*t3*a1*t1^2-2*t2^2*a0*t3*a2+2*t2*t3^2*t1*a0*a2+a2*t1^2*t3+t2^2*t3*a2+t2^2*t3*a3-a1*t2^2*t3+t3*a3*t1^2-a1*t1^2*t3-t3*a3-a2*t3+a1*t3-t2^2*t3*a2*t1^2+a1*t1^2*t2^2*t3-t2^2*t3*a3*t1^2+a1*a0*t1^2*t2^2*t3-t3*t1^2*a0*a2*t2^2-t3*t1^2*a0*a3*t2^2-a1*a0*t1*t2^3+t2^3*a0*t1*a2+t2^3*a0*t1*a3+t1*a0*t2*a2-t1*a0*t2*a3+t1*a0*t2*a1)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)+(1/2)*(2*a2+t3^2*a3-a3*t1^2-a2*t3^2-a2*t1^2-t3^2*a1+a1*t1^2)*dt2/(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2+(1/2)*(2*a2*a0*t1*t2^2-t1*a1*t3^2-a2*t1*t3^2+a1*a0*t1*t2^2*t3^2-a0*a3*t1*t2^2*t3^2+a2*a0*t1*t2^2*t3^2-2*a2*a0*t1^2*t2*t3-a3*t1+a2*t1+t1*a1+a0*a3*t2^3*t3-a2*a0*t2^3*t3-a1*a0*t2^3*t3+a3*t1*t3^2-a3*t1*t2^2*t3^2+a2*t1*t2^2*t3^2+t3^2*a0*t1*a2+t3^2*a0*t1*a3-t3^2*a0*t1*a1-a2*t1*t2^2-t1*a1*t2^2+a3*t1*t2^2+a1*t1*t2^2*t3^2-a0*a3*t2*t3-a0*a2*t2*t3+t2*a0*t3*a1)*dt3/((t3-1)*(t3+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2):

M[8, 4]:= -(1/2)*(a0*t2*a2*t1^2-a0*t2*a3*t1^2-a0*t2*a1*t1^2+2*t3*t1*t2^2*a0*a3-a3*a0*t1^2*t2*t3^2+a1*a0*t1^2*t2*t3^2-a2*a0*t1^2*t2*t3^2+t2*a2*t3^2-t2*a1*t3^2+a1*t1^2*t2*t3^2-a2*t1^2*t2*t3^2+t2*a3*t1^2+t2*a2*t1^2-t2*a1*t1^2-a3*t1^2*t2*t3^2-t2*a2-t2*a3+t2*a1+t2*a3*t3^2-2*t3^2*a0*t2*a3+a1*a0*t1*t3-a1*a0*t1*t3^3+a0*a3*t1*t3+a0*a3*t1*t3^3-a2*a0*t1*t3+a2*a0*t1*t3^3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)+(1/2)*(a0*a3*t1*t2^2+a2*a0*t1*t2^2-a1*a0*t1*t2^2-t1*a1*t3^2+a2*t1*t3^2+a1*a0*t1*t2^2*t3^2+a0*a3*t1*t2^2*t3^2-2*a0*a3*t1^2*t2*t3-a2*a0*t1*t2^2*t3^2+a3*t1-a2*t1+t1*a1-a3*t1*t3^2+a3*t1*t2^2*t3^2-a2*t1*t2^2*t3^2+2*t3^2*a0*t1*a3+a2*t1*t2^2-t1*a1*t2^2-a3*t1*t2^2+a1*t1*t2^2*t3^2-a0*a3*t2*t3-a0*a3*t2*t3^3-a0*a2*t2*t3+a0*a2*t2*t3^3-a1*a0*t2*t3^3+t2*a0*t3*a1)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)+(1/2)*(-a2*t1^2-a3*t1^2+a1*t1^2+2*a3-a3*t2^2+a2*t2^2-a1*t2^2)*dt3/(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2:

M[8, 5]:= -(1/2)*(-t2*a3*t1^2+t2*a2*t1^2-t2*a1*t1^2+2*t3*a0*t1+2*a1*t1*t3-t2*a1-t2*a2-2*a0*t2+t2*a3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))+(1/2)*(-a3*t1+a2*t1+t1*a1+2*a0*t1-2*a2*t2*t3-2*t2*a0*t3+a3*t1*t2^2+a2*t1*t2^2-t1*a1*t2^2)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(1/2)*(2-2*t2^2*t3^2*t1^2-a1-a2+2*t2^2*t1^2+2*t3^2*t2^2+2*t3^2*t1^2+a3-2*t1^2-2*t2^2-2*t3^2-2*a0*t1^2*t3^2-2*t2^2*t3^2*a0-2*a0^2*t2^2*t1^2-2*t2^2*a0^2*t3^2-2*t1^2*a0^2*t3^2+2*a0*t3^2-2*a0*t3^4+2*a3*t2^3*t1*t3-2*a1*t1*t2*t3+2*a1*t1*t2*t3^3+2*a3*t1^3*t2*t3+4*a3*t1*t2*t3^3-4*a3*t1^2*t2^2*t3^2-2*a2*t1*t2*t3+2*a2*t2*t3^3*t1-4*t2^2*a0*t1^2*t3^2+2*t3*t2^3*a0*t1+2*t1^3*t2*a0*t3-2*a0^2*t1^2*t2^2*t3^2+2*t1*t2*t3^3*a0^2+2*t2*a0^2*t1*t3+2*a0^2*t1*t2^3*t3-a3*t2^2+a1*t2^2+a2*t2^2-a3*t1^2+a1*t1^2+a2*t1^2+2*t3^2*a1-a1*t3^4+2*a2*t3^2-a2*t3^4-a3*t3^4-t3^2*a1*t1^2-a2*t3^2*t1^2-a3*t1^2*t3^2-t3^2*a1*t2^2-a2*t3^2*t2^2-a3*t2^2*t3^2+2*a0^2*t1^3*t2*t3+6*a0*t1*t2*t3^3-2*t3*a0*t1*t2)*dt3/((t3-1)*(t3+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2):

M[8, 6]:= -(1/2)*(t3*a3*t1^2-a1*t1^2*t3-a2*t1^2*t3+2*a1*t1*t2+2*t2*a0*t1-2*a0*t3-t3*a3-a1*t3+a2*t3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(1/2)*(2-2*t2^2*t3^2*t1^2-a1+a2+2*t2^2*t1^2+2*t3^2*t2^2+2*t3^2*t1^2-a3-2*t1^2-2*t2^2-2*t3^2-2*t2^2*t3^2*a0-2*a0^2*t2^2*t1^2-2*t2^2*a0^2*t3^2-2*t1^2*a0^2*t3^2+2*a0*t2^2+2*a3*t2^3*t1*t3-2*a1*t1*t2*t3+2*a1*t1*t2^3*t3-2*t3*t1*t2*a3+4*a2*t3*t1*t2^3-4*a2*t2^2*t3^2*t1^2+2*a2*t1^3*t2*t3+2*a2*t2*t3^3*t1-4*t2^2*a0*t1^2*t3^2+6*t3*t2^3*a0*t1+2*t1^3*t2*a0*t3-2*a0^2*t1^2*t2^2*t3^2+2*t1*t2*t3^3*a0^2+2*t2*a0^2*t1*t3+2*a0^2*t1*t2^3*t3-a3*t2^4+2*a3*t2^2+2*a1*t2^2+a3*t1^2+a1*t1^2-a1*t2^4-a2*t1^2-2*a0*t2^4-a2*t2^4+t3^2*a3+t3^2*a1-a2*t3^2-2*a0*t1^2*t2^2-a1*t2^2*t1^2-a2*t2^2*t1^2-t3^2*a1*t2^2-a2*t3^2*t2^2-a3*t2^2*t1^2-a3*t2^2*t3^2+2*a0^2*t1^3*t2*t3+2*a0*t1*t2*t3^3-2*t3*a0*t1*t2)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)+(1/2)*(-a2*t1+a3*t1+a3*t1*t3^2+2*a0*t1+a2*t1*t3^2+t1*a1-t1*a1*t3^2-2*a3*t2*t3-2*t2*a0*t3)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[8, 7]:= -(1/2)*(2-2*t2^2*t3^2*t1^2+a1-a2+2*t2^2*t1^2+2*t3^2*t2^2+2*t3^2*t1^2-a3-2*t1^2-2*t2^2-2*t3^2-2*a0*t1^2*t3^2-2*a0^2*t2^2*t1^2-2*t2^2*a0^2*t3^2-2*t1^2*a0^2*t3^2+2*a0*t1^2-4*t3^2*a1*t2^2*t1^2+2*a1*t1*t2*t3^3+2*a1*t1*t2^3*t3+4*a1*t1^3*t2*t3+2*a3*t1^3*t2*t3-2*t3*t1*t2*a3-2*a2*t1*t2*t3+2*a2*t1^3*t2*t3-4*t2^2*a0*t1^2*t3^2+2*t3*t2^3*a0*t1+6*t1^3*t2*a0*t3-2*a0^2*t1^2*t2^2*t3^2+2*t1*t2*t3^3*a0^2+2*t2*a0^2*t1*t3+2*a0^2*t1*t2^3*t3-2*a0*t1^4+a3*t2^2-a1*t2^2+a2*t2^2+2*a3*t1^2-a3*t1^4+2*a2*t1^2-a2*t1^4-a1*t1^4+t3^2*a3-t3^2*a1+a2*t3^2-2*a0*t1^2*t2^2-t3^2*a1*t1^2-a1*t2^2*t1^2-a2*t2^2*t1^2-a2*t3^2*t1^2-a3*t1^2*t3^2-a3*t2^2*t1^2+2*a0^2*t1^3*t2*t3+2*a0*t1*t2*t3^3-2*t3*a0*t1*t2)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2)^2)-(1/2)*(-t3*a3-a2*t3+a1*t3-2*a0*t3+2*a2*t1*t2+2*t2*a0*t1+t2^2*t3*a3-t2^2*t3*a2-a1*t2^2*t3)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(1/2)*(2*t3*a0*t1+2*a3*t1*t3-t2*a3-2*a0*t2-t2*a2+t2*a2*t3^2+t2*a1-t2*a3*t3^2-t2*a1*t3^2)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):

M[8, 8]:= (a0*t1^3-2*t1^3+2*t2*t3*t1^2+2*t1-t3^2*a0*t1-t2^2*a0*t1-a0*t1+2*t2*a0*t3-2*t2*t3)*dt1/((t1-1)*(t1+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(-a0*t2^3+2*t2^3-2*t3*t1*t2^2+a0*t3^2*t2+t2*a0*t1^2-2*t2+a0*t2+2*t3*t1-2*t3*a0*t1)*dt2/((t2-1)*(t2+1)*(t1^2-2*t1*t2*t3+t2^2-1+t3^2))-(-a0*t3^3+2*t3^3-2*t1*t2*t3^2-2*t3+a0*t2^2*t3+a0*t3+t3*a0*t1^2+2*t1*t2-2*t2*a0*t1)*dt3/((t1^2-2*t1*t2*t3+t2^2-1+t3^2)*(t3-1)*(t3+1)):