Example 9. Determine all triples of real numbers $(x, y, z)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
2 x+x^{2} y=y, \\
2 y+y^{2} z=z, \\
2 z+z^{2} x=x .
\end{array}\right.
$$
Solving by rearranging and factoring, the original system of equations can be transformed into
$$
\left\{\begin{array}{l}
y=\frac{2 x}{1-x^{2}}, \\
z=\frac{2 y}{1-y^{2}}, \\
x=\frac{2 z}{1-z^{2}} .
\end{array}\right.
$$

Let \( x = \tan \theta \left( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \right) \), then the original system of equations can be rewritten as
$$
\left\{\begin{array}{l}
y = \tan 2\theta, \\
z = \tan 4\theta, \\
x = \tan 8\theta .
\end{array}\right.
$$

Thus, \( \tan \theta = \tan 8\theta \).
Therefore, \( 8\theta = k\pi + \theta \), which simplifies to \( \theta = \frac{k\pi}{7} \) (where \( k \) is an integer).
Given \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \), we have \( -\frac{7}{2} < k < \frac{7}{2} \).
Hence, \( k = 0, \pm 1, \pm 2, \pm 3 \).
Therefore, the solutions to the system of equations are
$$
\begin{array}{l}
x = \tan \frac{k\pi}{7}, \quad y = \tan \frac{2k\pi}{7}, \quad z = \tan \frac{4k\pi}{7}. \\
(k = 0, \pm 1, \pm 2, \pm 3)
\end{array}
$$