Example $\mathbf{3}$ Find all real solutions of the system of three equations
$$
x\left(y^{2}+2 z^{2}\right)=y\left(z^{2}+2 x^{2}\right)=z\left(x^{2}+2 y^{2}\right)
$$

(63rd Czech and Slovak Mathematical Olympiad)
Solve:
$$
\begin{array}{l}
x\left(y^{2}+2 z^{2}\right)=y\left(z^{2}+2 x^{2}\right) \\
\Rightarrow(2 x-y)\left(z^{2}-x y\right)=0 \\
\Rightarrow y=2 x \text { or } z^{2}=x y .
\end{array}
$$
(1) When $y=2 x$, substitute into
$$
y\left(z^{2}+2 x^{2}\right)=z\left(x^{2}+2 y^{2}\right)
$$

we get $2 x\left(2 x^{2}+z^{2}\right)=9 x^{2} z$.
If $x=0$, then $y=0$, substituting into the original equation, $z$ is any real number, hence
$$
(x, y, z)=(0,0, t) \text { ( } t \text { is any real number) } \text {; }
$$

If $x \neq 0$, then
$$
\begin{array}{l}
4 x^{2}-9 x z+2 z^{2}=0 \\
\Rightarrow x=2 z \text { or } z=4 x \\
\Rightarrow(x, y, z)=(2 t, 4 t, t),(t, 2 t, 4 t)(t \text { is }
\end{array}
$$

any non-zero real number).
(2) When $z^{2}=x y$, substitute into
$$
x\left(y^{2}+2 z^{2}\right)=z\left(x^{2}+2 y^{2}\right)
$$

we get $x y(2 x+y)=z\left(x^{2}+2 y^{2}\right)$.
(i) If $x=y=0$, then
$(x, y, z)=(0,0, t)$ ( $t$ is any real number).
(ii) If only one of $x y$ is 0, then $z=0$, at this time the solution of the equation system is
$(x, y, z)$
$=(0, t, 0),(t, 0,0)$ ( $t$ is any non-zero real number $)$.
(iii) If $x y \neq 0$, then $z=\frac{x y(2 x+y)}{x^{2}+2 y^{2}}$.
Thus, $x y=z^{2}=\frac{x^{2} y^{2}(2 x+y)^{2}}{\left(x^{2}+2 y^{2}\right)^{2}}$.
Note that, $x y \neq 0, x y(2 x+y)^{2}=\left(x^{2}+2 y^{2}\right)^{2}$.
Rearranging gives $(4 y-x)(x-y)\left(x^{2}+x y+y^{2}\right)=0$.
But $x^{2}+x y+y^{2}=\left(x+\frac{y}{2}\right)^{2}+\frac{3}{4} y^{2}>0$, so
$x=4 y$ or $x=y$.
Thus, substituting into the original equation system, we get
$$
\begin{array}{l}
(x, y, z) \\
=(t, t, t),(4 t, t, 2 t)(t \text { is any non-zero real number }) \text {. } \\
\text { Hence }(x, y, z) \\
=(0,0,0),(0,0, t),(0, t, 0),(t, 0,0), \\
\quad(2 t, 4 t, t),(t, 2 t, 4 t),(4 t, t, 2 t),(t, t, t)
\end{array}
$$

Thus, $(x, y, z)$
( $t$ is any non-zero real number).