10 Find all integers $x, y$ such that $x^{2} + xy + y^{2} = 1$.

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10. Multiply both sides by 4, then complete the square, to get
$$(2 x+y)^{2}+3 y^{2}=4$$

Thus, $4-3 y^{2}$ is a perfect square, which requires $y^{2}=0$ or 1, corresponding to $(2 x+y)^{2}=4,1$. Solving these respectively yields
$$(x, y)=( \pm 1,0),(0, \pm 1),(1,-1),(-1,1)$$