What method should I use to solve this system of polynomial equations? What method should I use to solve this system of polynomial equations?
$$\begin{cases} 3x^3 - 3y^3 + z^3 - xyz - 3 &= 0\\
                  3y^3 -  x^3 - z^3 - xyz + 5 &= 0\\
                   x^3 -  y^3 + z^3 - xyz - 2 &= 0\end{cases}$$

I've run out of ideas. Tried adding, subtracting but it gave me nothing. Maybe there's a good substitution I don't see?

Hint. By summing the first two equations we obtain
$$xyz=x^3+1.$$
Then solve the linear system with respect to $y^3$ and $z^3$:
$$\begin{cases}
3x^3 - 3y^3 + z^3 - (x^3+1) - 3 = 0\\
x^3-y^3+z^3-(x^3+1)-2=0
\end{cases}$$
and we find
$$y^3=x^3-\frac{1}{2},\quad z^3=x^3+\frac{5}{2}.$$
Going back to the first equation cubed we get
$$x^3\left(x^3-\frac{1}{2}\right)\left(
x^3+\frac{5}{2}\right)=(x^3+1)^3.$$
Can you take it from here?
