4-175 Find all possible 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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［Solution］Let integers $x, y$ satisfy the given conditions, then we have
$$
7(x+y)=3\left(x^{2}-x y+y^{2}\right) .
$$

Let $p=x+y, q=x-y$,
then $\quad x=\frac{p+q}{2}, y=\frac{p-q}{2}$.

Substituting into (1) yields $28 p=3\left(p^{2}+3 q^{2}\right)$.
From this, we can deduce that $3 \mid p$, and $p$ is a non-negative integer. Let $p=3 k$, where $k$ is a non-negative integer, substituting into (2) gives
$$
28 k=3\left(3 k^{2}+q^{2}\right) .
$$

Similarly, $3 \mid k$. Let $k=3 m$ (where $m$ is a non-negative integer). Substituting into (3) gives
$$
28 m=27 m^{2}+q^{2} \text {, }
$$

which is $m(28-27 m)=q^{2}$.
Since $\quad q^{2} \geqslant 0$,
it follows that $m(28-27 m) \geqslant 0$, thus $m=0$ or $m=1$.
If $m=0$, then $p=q=0, x=y=0$ (discard).
If $m=1$, then $q= \pm 1, p=q, x=4, y=5$ or $x=5, y=4$.
Therefore, the pairs $(x, y)$ that satisfy the conditions are $(5,4)$ or $(4,5)$.