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?