Existence of the integral equation solution

Question

In the following equation:

$$\int_{-\infty}^\infty g(x) \frac{1}{\sqrt{2\pi\sigma^2}}e^{-(x-\mu)^2/(2\sigma^2)} dx = \frac{1}{\mu^2 - 1} , $$

The function $g(x)$ is unknown and doesn't depend on $\mu$.

Does the solution exist with $\mu>1$?