10. Find the integer solutions of the system of equations
$$
\left\{\begin{array}{l}
x+y+z=0, \\
x^{3}+y^{3}+z^{3}=-18
\end{array}\right. 
$$
$$
\begin{array}{l}
x^{3}+y^{3}+z^{3}-3 x y z=(x+y)^{3}+z^{3}-3 x y(x+y)-3 x y z \\
=(x+y+z)\left[(x+y)^{2}+z^{2}-(x+y) z\right]-3 x y(x+y+z) \\
=(x+y+z)\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\right)=0 \Rightarrow x^{3}+y^{3}+z^{3}=3 x y z, \\
\text { Thus }\left\{\begin{array}{l}
x+y+z=0, \\
x y z=-6
\end{array} \Rightarrow(x, y, z)=(1,2,-3),(1,-3,2),(2,1,-3),(2,-3,1),\right.
\end{array}
$$

Thus $\left\{\begin{array}{l}x+y+z=0, \\ x y z=-6\end{array} \Rightarrow(x, y, z)=(1,2,-3),(1,-3,2),(2,1,-3),(2,-3,1)\right.$, $(-3,1,2),(-3,2,1)$, a total of 6 solutions.