.
1. Right sides are identical ====> hence, left sides are equal:
x^3 = y^3.
2. Since x and y are integer, the last equality implies x = y.
3. Then the either of the two given equations is equivalent to
x^3 = 89x + 40x, or
x^3 = 129x,
x*(x^2-129) = 0.
In integer numbers, the last equation has only one solution x= 0.
Answer. The only solution to the original system in integer numbers is the pair (x,y) = (0,0).
