Example 1 Find the integer solutions of the equation
$$x y-10(x+y)=1$$
Solving by using the cross-multiplication method, we can transform (1) into
$$(x-10)(y-10)=101$$

Since 101 is a prime number, we have
$$\begin{aligned}
& (x-10, y-10) \\
= & (1,101),(101,1),(-1,-101),(-101,-1) .
\end{aligned}$$

Solving each case, we get the integer solutions of the equation as
$$(x, y)=(11,111),(111,11),(9,-91),(-91,9)$$