18.3.41 Find all integer solutions to the indeterminate equation
$$
x^{3}+y^{3}+z^{3}=3 x y z
$$
Let $\left(x_{1}, y_{1}, z_{1}\right)$ be a set of integer solutions to (1), then
$$
\begin{array}{c}
x_{1}^{3}+y_{1}^{3}+z_{1}^{3}-3 x_{1} y_{1} z_{1}=\left(x_{1}+y_{1}+z_{1}\right)\left(x_{1}^{2}+y_{1}^{2}\right. \\
\left.+z_{1}^{2}-x_{1} y_{1}-x_{1} z_{1}-y_{1} z_{1}\right)=0 . \\
x_{1}+y_{1}+z_{1}=0
\end{array}
$$

Thus, we have
or
$$
x_{1}^{2}+y_{1}^{2}+z_{1}^{2}-x_{1} y_{1}-x_{1} z_{1}-y_{1} z_{1}=0
$$

From (3), we get
$$
\begin{array}{c}
\left(x_{1}-y_{1}\right)^{2}+\left(y_{1}-z_{1}\right)^{2}+\left(x_{1}-z_{1}\right)^{2}=0 \\
x_{1}=y_{1}=z_{1} .
\end{array}
$$

Therefore, from (2) and (4), the complete set of positive integer solutions to equation (1) are:
(1) $x=y=z=k$,
(2) $x=m, y=n, z=-m-n$,
(3) $x=m, y=-m-n, z=n$, (4) $x=-m-n, y=m, z=n$, where $k, m, n$ are any integers.