Example 3 Find all integer solutions to $111 x-321 y=75$.

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Since
$$111=3 \times 37, \quad 321=3 \times 107$$
$$75=3 \times 25$$ and $$111 x-321 y=75$$, we get
$$37 x-107 y=25$$

Thus, the solutions to $$111 x-321 y=75$$ are exactly the same as those to equation (13). Now, let's solve
$$37 s+107 t=1$$

Since
$$107=2 \times 37+33, \quad 37=33+4, \quad 33=8 \times 4+1$$

We have
$$\begin{aligned}
1= & 33-8 \times 4=37-4-8 \times 4=37-9 \times 4 \\
& =37-9 \times(37-33)=9 \times 33-8 \times 37 \\
& =9 \times(107-2 \times 37)-8 \times 37=9 \times 107 \\
& -26 \times 37=37 \times(-26)+107 \times 9
\end{aligned}$$

Thus, a set of integer solutions to equation (14) is $$s=-26, t=9$$. Let $$x=25 s$$, $$y=-25 t$$, then from equation (14) we have
$$37 x-107 y=25(37 s+107 t)=25$$

Therefore, $$x=25 \times(-26)=-650, y=-25 \times 9=-225$$ is a set of integer solutions to equation (13). By Theorem 1, all integer solutions to equation (13) can be expressed as
$$x=-650+107 t, \quad y=-225+37 t$$

or
$$x=-8+107 t, \quad y=-3+37 t$$
(where $$t=0, \pm 1, \pm 2, \pm 3, \cdots$$).