1. Find all integer values of $x, y$ that satisfy the condition $\frac{x+y}{x^{2}-x y+y^{2}}=\frac{3}{7}$.

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1. The original equation is transformed into $3 x^{2}-(3 y+7) x+3 y^{2}-7 y=0$. From the condition for the quadratic equation in $x$ to have solutions $\Delta \geqslant 0$ and $y$ being an integer, we get $0 \leqslant y \leqslant 5$, i.e., $y=0,1,2,3,4,5$. Substituting each value into the original equation, we find that the original equation has only two solutions: $(4,5),(5,4)$.