In the book Quantum mechanics by Eugen Merzbacher, third edition, at page 462 he claims that this differential equation (for the unknown operator $F_0=F_0(x,y,z)$) can be solved by separation of variables:
$$[F_0,H_0]|1s\rangle=eEz|1s\rangle,$$
where $H_0$ is the non-relativistic Hamiltonian for the hydrogen atom:
$$H_0=-\frac{\hbar^2}{2m}\nabla^2-\frac{e^2}{4\pi\varepsilon_0}\frac{1}{r},$$
$|1s\rangle$ is the fundamental state of the hydrogen atom:
$$|1s\rangle=\frac{1}{\sqrt{\pi a_0^3}}\exp(-r/a_0),$$
$r=\sqrt{x^2+y^2+z^2}$, $[F_0,H_0]=F_0H_0-H_0F_0$ and the other are constants.
So I tried to use separation of variables $F_0(x,y,z)=X_0(x)Y_0(y)Z_0(z)$, and the equation that I reached is:
$$\frac{X_0Y_0Z_0}{a_0^2r}(r-2a_0)+X_0^{\prime\prime}Y_0Z_0+X_0Y_0^{\prime\prime}Z_0+X_0Y_0Z_0^{\prime\prime}=-\frac{2m}{\hbar^2}eEz$$
and I can't proceed from here, it's different from, for example, the wave equation or the heat equation where you can separate the capital functions on each side of the equation. I also tried $F_0(r,z)=R_0(r)Z_0(z)$, but couldn't proceed too.
My questions are: does this equation can really be solved by separation of variables? If yes, what am I doing wrong? If no, what would you suggest me?
EDIT: The solution given by the author is
$$F_0=-\frac{eEma_0}{\hbar^2}\left( \frac{r}{2}+a_0\right )z$$
One thing that I'm supposing to be true is that, once $F_0$ is a function of the coordinates only, it commutes with any other function of the coordinates only. With this the potential energy term vanish.