Solution:
Answer: $(-670, -670)$, $(1005, -335)$, $(-335, 1005)$.

To prove this, let $z = -670$. We have
$$0 = x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2}(x + y + z)\left((x - y)^{2} + (y - z)^{2} + (z - x)^{2}\right)$$
Thus either $x + y + z = 0$, or $x = y = z$. In the latter case we get $x = y = -670$, which satisfies both the equations.

In the former case we get $x + y = 670$. Then at least one of $x, y$ is positive, but not both, as from the second equation we would get $x + y = 1340$. If $x > 0 \geq y$, we get $x - y = 1340$, which together with $x + y = 670$ yields $x = 1005$, $y = -335$. If $y > 0 \geq x$ we get similarly $x = -335$, $y = 1005$.