I just had a similar problem and also asked here for help. In my case it finally turned out that the processor got too hot. Running your code on my computer also heated up the processor considerably.

So, my suggestion is to monitor the processor temperature during your computation and compare it with computations that don't stop your computer. If it then looks like a temperature problem, you will find ways to solve it. I have no experience with the matter.

Thank you!! I will keep it in mind!

Dear Claudio, I still think it is solvable.

 p1*a1*z1 + p2*a2*z2 == z3*P01 && p1*z1 == z3*P00

Above I denote the terms in brackets of the initial system as z1, z2 and z3 respectively. This linear system has a solution for z1/z3 and z2/z3, provided z3<>0. This solution will not contain x0 or x1 as arguments, they will be all inside z1/z3 and z2/z3. But then one can just invert the power (given that it is an integer and that the roots are non-negative). Then one can just solve the resulting system, which will be linear in x0 and x1, just a bit ugly.

The question is, why does Mathematica fail here. I do not think I am smarter then my computer after all...