6.16*. (GDR, 80). Solve the system

$$
\left\{\begin{array}{l}
2 x+x^{2} y=y \\
2 y+y^{2} z=z \\
2 z+z^{2} x=x
\end{array}\right.
$$
6.16. Let the set ( $x ; y ; z$ ) satisfy the original system. If we denote $x=\operatorname{tg} a$, where $a \in(-\pi / 2 ; \pi / 2)$, then from the first equation of the system we get

$$
y=2 x^{\prime}\left(1-x^{2}\right)=\operatorname{tg} 2 a
$$

from the second-

$$
z=\operatorname{tg} 4 a
$$

and from the third-

$$
x=\operatorname{tg} 8 a
$$

Hence

$$
\operatorname{tg} a=\operatorname{tg} 8 a, \quad \text { i.e. } \quad 7 a=\pi k \quad(k \in \mathbb{Z})
$$

Therefore, $a=\pi k / 7$. In this case

$$
x=\operatorname{tg}(\pi k / 7), \quad y=\operatorname{tg}(2 \pi k / 7), \quad z=\operatorname{tg}(4 \pi k / 7)
$$

Verification shows that for $k \in\{-3 ;-2 ;-1 ; 0 ; 1 ; 2 ; 3\}$, the 7 sets of numbers $x, y, z$ are distinct and indeed satisfy the original system.