1. Find all integers $x, y$ that satisfy the equation
$$
x+y=x^{2}-x y+y^{2} \text {. }
$$
1. Solving for $y$, we get:
$$
y=\frac{x+1 \pm \sqrt{-3(x-1)^{2}+4}}{2} .
$$

Thus $3(x-1)^{2} \leq 4$, i.e.,
$$
1-\frac{2}{\sqrt{3}} \leq x \leq 1+\frac{2}{\sqrt{3}} \text {. }
$$

Thus $x=0,1,2$ and $(x, y)=(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)$ are all the solutions.