Example 1 Find the integer solutions of the equation $x+y=x^{2}-x y+y^{2}$.
Analysis: The highest degree of this equation is 2, consider whether it can be handled using a quadratic equation.
Solution: The original equation is rearranged as $x^{2}-(1+y) x+y^{2}-y=0$. For the equation to have integer solutions, it must be true that
$$
\Delta=(y+1)^{2}-4\left(y^{2}-y\right) \geqslant 0 \text {, solving this yields } \frac{3-2 \sqrt{3}}{3} \leqslant y \leqslant \frac{3+2 \sqrt{3}}{3} \text {, }
$$

The integers satisfying this inequality are only $y=0,1,2$.
When $y=0$, the original equation gives $x=0$ or $x=1$;
When $y=1$, the original equation gives $x=2$ or 0;
When $y=2$, the original equation gives $x=1$ or 2.
Therefore, the integer solutions of the equation are $\left\{\begin{array}{l}x=0 \\ y=0\end{array},\left\{\begin{array}{l}x=1 \\ y=0\end{array},\left\{\begin{array}{l}x=2 \\ y=1\end{array},\left\{\begin{array}{l}x=0 \\ y=1\end{array},\left\{\begin{array}{l}x=1 \\ y=2\end{array},\left\{\begin{array}{l}x=2 \\ y=2\end{array}\right.\right.\right.\right.\right.\right.$.