$4 \cdot 239$ If integers $x, y$ satisfy $y^{2}+3 x^{2} y^{2}=30 x^{2}+517$, find $3 x^{2} y^{2}$.
[Solution] The original equation can be rewritten as
$$
\left(3 x^{2}+1\right)\left(y^{2}-10\right)=507=3 \times 13^{2}.
$$

Since $3 x^{2}+1$ is coprime with 3, it can only be 1, 13, or $13^{2}$. Correspondingly, $y^{2}-10$ would be $3 \times 13^{2}$, $3 \times 13$, or 3. However, only when $y^{2}-10=3 \times 13$, $y^{2}=49$ is a perfect square, and the corresponding $3 x^{2}=13-1=12$. Therefore,
$$
3 x^{2} y^{2}=12 \times 49=588
$$