8. If integers $a, b, c$ satisfy:
$$
a+b+c=3, a^{3}+b^{3}+c^{3}=3 \text {, }
$$

then the maximum value of $a^{2}+b^{2}+c^{2}$ is $\qquad$
8.57.

From the problem, we know
$$
a^{2} b+a b^{2}-3 a^{2}-3 b^{2}-6 a b+9 a+9 b-8=0 \text {. }
$$

Considering it as a quadratic equation in $a$, we rearrange it to get
$$
(b-3) a^{2}+\left(b^{2}-6 b+9\right) a-3 b^{2}+9 b-8=0 \text {. }
$$

Since $a$ is an integer, 
$$
\Delta=(b-1)^{2}(b-3)(b+5)
$$

must be a perfect square.
If $\Delta=0$, then $b=1,3,-5$;
If $\Delta \neq 0$, then $b=4$.
Thus, $(a, b, c)$
$$
\begin{aligned}
= & (1,1,1),(4,-5,4),(4,4,-5), \\
& (-5,4,4) .
\end{aligned}
$$

Therefore, the maximum value of $a^{2}+b^{2}+c^{2}$ is 57.