1.1. Find all integers $m$ for which the equation
$$
x^{3}-m x^{2}+m x-\left(m^{2}+1\right)=0
$$
has an integer solution.
1.1. Suppose $p$ is an integer such that $p^{3}-m p^{2}+m p-\left(m^{2}+1\right)=0$. It follows that $\left(p^{2}+m\right)(p-m)=1$. Since $p$ and $m$ are integers, we have either
(1) $p^{2}+m=p-m=-1$, or
(2) $p^{2}+m=p-m=1$.

In case (1), we have $m=p+1$, and so $p^{2}+p+1=-1$ or $p^{2}+p+2=0$ which has no real solution.
In case (2), we have $m=p-1$, and so $p^{2}+p-1=1$ or $p^{2}+p-2=0$ which has the solutions $p=-2$ and 1 .
Hence, $m=-3$ and 0 are the integer values of $m$ for which the given equation has an integer solution.