Example 1 Find all integer solutions to the indeterminate equation $x^{2}=y^{2}+2 y+13$.
(46th Moscow Mathematical Olympiad)
The original equation can be transformed into
$$
x^{2}-(y+1)^{2}=12 \text {. }
$$

By the difference of squares formula, we get
$$
(x+y+1)(x-y-1)=12 \text {. }
$$

Since \(x+y+1\) and \(x-y-1\) have the same parity, and 12 is an even number, we have
$$
\begin{array}{l}
(x+y+1, x-y-1) \\
=(2,6),(6,2),(-2,-6),(-6,-2) \\
\Rightarrow(x, y)=(4,-3),(4,1),(-4,1), \\
(-4,-3) .
\end{array}
$$