13. Given that $a, b$ are real numbers, the system of equations about $x, y$
$$
\left\{\begin{array}{l}
y=x^{3}-a x^{2}-b x, \\
y=a x+b
\end{array}\right.
$$

has integer solutions $(x, y)$. Find the relationship that $a, b$ must satisfy.
13. From the given, we have $y=x^{3}-x y$, which means $(x+1) y=x^{3}$.
It is clear that $x+1 \neq 0$, thus,
$$
y=\frac{x^{3}}{x+1}=x^{2}-x+1-\frac{1}{x+1} \text {. }
$$

Since $x, y$ are both integers, we have,
$$
x+1= \pm 1 \text {. }
$$

Therefore, $x=-2$ or $x=0$.
Thus, $y=8$ or $y=0$.
Hence, $x=-2, y=8$ or $x=0, y=0$.
When $x=-2, y=8$, substituting into $y=a x+b$, we get
$$
2 a-b+8=0 \text {; }
$$

When $x=0, y=0$, substituting into $y=a x+b$, we get $b=0$.
In summary, the relationship that $a$ and $b$ satisfy is
$2 a-b+8=0$ or $b=0$ (where $a$ is any real number).