Kashyap's system of equations

How many ordered triples of integer solutions (x,y,z)(x,y,z) are there to the following system of equations:

x+y=1z,x3+y3=1z2,z1?\begin{aligned} x + y & = 1 - z, \\ x^3 + y^3 & = 1 - z^2,\\ z & \ne 1 ? \end{aligned}

This problem is shared by Kashyap M.

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