DSolve unevaluated. System of differential equations

I am trying to solve a system of differential equations with partial derivatives. The initial equation is just one

$$
frac{(v-1)^3}{(a(u-1) u v +b (v-1))^2} Big(frac{partial u}{partial x} frac{partial v}{partial y} – frac{partial v}{partial x} frac{partial u}{partial y} Big)
=
2 frac{x^3 y^3 (x-1)}{(a(x-1) (y-1) + b x (1-xy))^2}
$$

But since a and b are independent, this can be rearranged into 3 different equations: passing the denominators multiplying to each side, and finding the coefficient of $a^2$, $b^2$, and $a b$, we find the three different equations. However, when I feed it to DSolve, the input stays unevaluated… Perhaps Mathematica cannot solve this system?

eq1 = 2*(-1 + x)*x^3*y^3*(-1 + v(x, y))^2 == 
  x^2*(-1 + x*y)^2*(-1 + v(x, y))^3*(Derivative(0, 1)(v)(x, y)*
      Derivative(1, 0)(u)(x, y) - 
     Derivative(0, 1)(u)(x, y)*Derivative(1, 0)(v)(x, y))

eq2 = 2*(-1 + x)*x^3*y^3*(-1 + u(x, y))^2*u(x, y)^2*
   v(x, y)^2 == (-1 + x)^2*(-1 + y)^2*(-1 + 
      v(x, y))^3*(Derivative(0, 1)(v)(x, y)*
      Derivative(1, 0)(u)(x, y) - 
     Derivative(0, 1)(u)(x, y)*Derivative(1, 0)(v)(x, y))

eq3 = 4*(-1 + x)*x^3*y^3*(-1 + u(x, y))*u(x, y)*(-1 + v(x, y))*
   v(x, y) == -2*(-1 + x)*
   x*(-1 + y)*(-1 + 
     x*y)*(-1 + v(x, y))^3*(Derivative(0, 1)(v)(x, y)*
      Derivative(1, 0)(u)(x, y) - 
     Derivative(0, 1)(u)(x, y)*Derivative(1, 0)(v)(x, y))


DSolve({eq1, eq2, eq3}, {u(x, y), v(x, y)}, {x, y})

Any help or suggestion would be much appreciated