Example 6 Find all integer solutions of $\left(x^{2}-y^{2}\right)^{2}=1+16 y$.

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Solve: Since the left side of the equation $\geqslant 0$, the right side $\geqslant 0$, thus $y \geqslant 0$. It is also clear that $x^{2}-y^{2} \neq 0$, and since $x_{1} y$ is an integer, we have $|x| \geqslant y+1$, or $|x| \leqslant y-1$.

When $|x| \geqslant y+1$, the left side of the equation $\geqslant\left((y+1)^{2}-y^{2}\right)^{2}=(2 y+1)^{2}$.
When $|x| \leqslant y-1$, at this time $y-1 \geqslant 0$, and $y^{2}-x^{2} \geqslant y^{2}-(y-1)^{2}=2 y-$ $1>0$, so the left side of the equation $\geqslant(2 y-1)^{2}$.

Therefore, from the original equation, we get
$$(2 y-1)^{2} \leqslant 1+16 y$$

Thus, $0 \leqslant y \leqslant 5$. Checking each value, we find all integer solutions to be $\left(x_{1} y\right)=( \pm 1,0),( \pm 4,3)$, $( \pm 4,5)$.