Example 3 Find the circle passing through the intersection points of the circles $x^{2}+y^{2}+4 x-3=0$ and $x^{2}+y^{2}-4 y-3=0$, and whose center lies on the line $2 x-y-$ $4=0$.
Let the equation of the required circle be $x^{2}+y^{2}+4 x-3+\lambda\left(x^{2}+y^{2}-4 y-3\right)=0$,
i.e., $(1+\lambda) x^{2}+(1+\lambda) y^{2}+4 x-4 \lambda y-3-3 \lambda=0$,
the center of the circle is $\left(-\frac{2}{1+\lambda}, \frac{2 \lambda}{1+\lambda}\right)$. It lies on the given line $2 x-y-4=0$, $-\frac{4}{1+\lambda}-\frac{2 \lambda}{1+\lambda}-4=0$, $\lambda=-\frac{4}{3}$, thus the required equation is: $x^{2}+y^{2}-12 x-16 y-3=0$.