11.3. Find all solutions of the system of equations in real numbers:

$$
\left\{\begin{array}{l}
x^{5}=y^{3}+2 z \\
y^{5}=z^{3}+2 x \\
z^{5}=x^{3}+2 y
\end{array}\right.
$$
Answer. Three solutions: $(0,0,0), \pm(\sqrt{2}, \sqrt{2}, \sqrt{2})$.

Solution. 1. Let's find the easily guessable solutions of the system, when all variable values are equal to each other, that is, when $x=y=z$. Then $x^{5}-x^{3}-2 x=0$, from which $x=0$ or $x= \pm \sqrt{2}$. This leads to the three solutions indicated in the answer: $(0,0,0), \pm(\sqrt{2}, \sqrt{2}, \sqrt{2})$. We will prove that there are no other solutions.