I am looking for the steady state solution of a Fokker-Planck equation. The process involves a constant drift and position-dependent removal/insertion, thus leading to non-zero a steady state probability current. The equation for the steady state probability distribution $P(x)$ reads \begin{equation} \left(v\partial_x + D \partial^2_x \right) P(x) = \gamma(x)P(x)-\gamma(-x)P(-x) \end{equation} where $\gamma(x)$ is an arbitrary function for which I only assume $\gamma(x) > 0$ and $\partial_x\gamma(x) > 0$, which allows for the existence of a non-trivial solution. I have been trying to solve this equation using an integrating factor but without luck. It might be that this is fairly trivial but I am currently stuck. Thank you in advance for your help.
P.S. I would like to keep $\gamma(x)$ as general as possible in order to later perform some functional derivative with respect to it.