5. Find all real solutions of the system of equations
$$
\left\{\begin{array}{l}
(x+1)\left(x^{2}+1\right)=y^{3}+1 \\
(y+1)\left(y^{2}+1\right)=z^{3}+1 \\
(z+1)\left(z^{2}+1\right)=x^{3}+1
\end{array}\right.
$$
5. Consider the polynomials $x^{3}+x^{2}+x$ and $x^{3}$ on the intervals $(0,+\infty)$, $(-\infty,-1)$, and $(-1,0)$.

(1) If $x>0$, then $x^{3}+x^{2}+x>x^{3}>0$.
From the given,
$$
\begin{array}{l}
y^{3}>x^{3} \Rightarrow y>x>0 \\
\Rightarrow z>y>0, x>z>0 .
\end{array}
$$

Contradiction.
(2) If $x<0$, then $x^{3}+x^{2}+x<0$. Hence $-1>x^{3}+x^{2}+x>x^{3}$.
From the given,
$$
-1>y>x \Rightarrow-1>z>y,-1>x>z \text {. }
$$

Contradiction.
(3) If $x \in(-1,0)$, then $x^{3}+x^{2}>0$ and $x^{2}+xy>x \Rightarrow-1>z>y,-1>x>z \text {. }
$$

Contradiction.
When $x=0,-1$, the solutions to the system of equations are
$$
(x, y, z)=(0,0,0) \text { and }(-1,-1,-1) \text {, }
$$

which can be easily verified to satisfy the problem's requirements.