Example 7. Find the integer solutions of $x^{2}-6 y^{2}=1$.

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Solve $x^{2}=1+6 y^{2}$, substituting $y=1,2,3, \cdots$ to make $1+6 y^{2}$ a perfect square, the smallest positive value is $y=2$, yielding the particular solution $\left\{\begin{array}{l}x=5, \\ y=2 \text {. }\end{array}\right.$ Then the other integer solutions can be given by $x+y \sqrt{6}= \pm(5 +2 \sqrt{6})^{n}$. For example, $n=2, x+y \sqrt{6}$ $= \pm(49+20 \sqrt{6})$, yielding $\left\{\begin{array}{l}x=49,-49, \\ y=20,-20 .\end{array}\right.$ The rest can be deduced similarly.

For equations of the form $x^{2}-D y^{2}=-1$, if $D$ is an integer and not a perfect square, similarly use the trial solution method to find $a, b$ as the smallest positive integer solutions, then the complete integer solutions of the equation are determined by $x+y \sqrt{D}= \pm(a+b \sqrt{D})^{n+1}$ (where $n$ is any integer). Due to space limitations, examples are not provided here.