Example 4 Find all integer solutions to the equation $\frac{x+y}{x^{2}-x y+y^{2}}=\frac{3}{7}$.
(12th All-Russian Mathematical Olympiad)
Notice that
$$
\begin{array}{l}
x^{2}-x y+y^{2} \\
=\frac{1}{4}\left[(x+y)^{2}+3(x-y)^{2}\right] .
\end{array}
$$

Let \( x+y=p, x-y=q \) (where \( p, q \in \mathbf{Z} \)), then the original equation becomes \( 28 p=3\left(p^{2}+3 q^{2}\right) \).
From this, we know that \( p>0 \), and \( p \) is a multiple of 3.
Let \( p=3 k \) (where \( k \in \mathbf{N}_{+} \)), then
$$
28 k=3\left(3 k^{2}+q^{2}\right) \text {. }
$$

Thus, \( k \) is a multiple of 3.
Let \( k=3 m \) (where \( m \in \mathbf{N}_{+} \)), then
$$
28 m=27 m^{2}+q^{2} \text {. }
$$

From \( 28 m-27 m^{2}=q^{2} \geqslant 0 \), we know that \( m=1 \).
At this point, \( k=3, p=9, q= \pm 1 \).
From \( x+y=9, x-y= \pm 1 \), we solve to get the integer solutions of the original equation as \( (x, y)=(5,4),(4,5) \).