Example 2 Find the integer solutions of the equation $\frac{1}{x^{2}}+\frac{1}{x y}+\frac{1}{y^{2}}=1$.

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Solution: The original equation can be transformed into $y^{2}+x y+x^{2}=x^{2} y^{2}$. Multiplying both sides of the equation by 4, and then adding $4 x y+1$, we get $4(x+y)^{2}+1=(2 x y+1)^{2}$.

Let $p=2(x+y), q=(2 x y+1)$, then the original equation can be transformed into $p^{2}+1=q^{2}$, which is $(q+p)(q-p)=1$. Therefore,
$\left\{\begin{array}{l}q=1 \\ p=0\end{array}\right.$ or $\left\{\begin{array}{l}q=-1 \\ p=0\end{array}\right.$, which leads to $\left\{\begin{array}{l}x= \pm 1 \\ y= \pm 1\end{array}\right.$

Thus, the integer solutions of the original equation are $(1,-1),(-1,1)$.