Example 3 Try to find the integer solutions of the equation $x^{2}-23 x y^{2}+1989 y^{2}=0$.

 untranslated part:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 

The above text has been translated into English, preserving the original text's line breaks and format, as requested.
Obviously, $x=y=0$ is a solution to the equation, and if one of the unknowns is 0, then the other unknown is also 0.

Assume $x \neq 0$ and $y \neq 0$. Clearly, $y^{2} \mid x^{2}$, so we can let $x=t y, t \in \mathbf{Z}$, substituting into the equation we get
$$
t^{2}-23 y t+1989=0 \text {. }
$$

By Vieta's formulas, we have
$$
\begin{array}{l}
t_{1}+t_{2}=23 y, \\
t_{1} t_{2}=1989=3^{2} \times 13 \times 17 .
\end{array}
$$

Solving, we get $\left\{\begin{array}{l}t_{1}=9, \\ t_{2}=221 ;\end{array}\left\{\begin{array}{l}t_{1}=221, \\ t_{2}=9 ;\end{array}\right.\right.$
$$
\left\{\begin{array} { l } 
{ t _ { 1 } = - 9 , } \\
{ t _ { 2 } = - 2 2 1 ; }
\end{array} \left\{\begin{array}{l}
t_{1}=-221, \\
t_{2}=-9 .
\end{array}\right.\right.
$$

Thus, $y= \pm 10$.
Therefore, the integer solutions are $(x, y)=(90,10)$,
$$
(2210,10),(90,-10),(2210,-10) \text {. }
$$