Example 3 Find the integer solutions of the equation $x^{2}-5 x y+6 y^{2}-3 x+5 y-$ $25=0$.
Solution: The equation is $(x-2 y+1)(x-3 y-4)=21$. Solving the system of equations, we get the solutions as $(x, y)=(-50,-25)$,
$$
\begin{array}{l}
(4,-1),(-26,-9),(-16,-9),(-6, \\
-1),(50,15),(-72,-25) .
\end{array}
$$