In Kerr background given by Boyer-Lindquist(BL) coordinates ($t,\, r,\, \theta,\, \phi $):
$$ g_{\mu\nu} dx^{\mu}dx^{\nu}=-(1-\frac{2Mr}{r^2+a^2\cos^2{\theta}})\,dt^2-\frac{r^2+a^2\cos^2{\theta}}{r^2+2Mr +a^2} dr^2 +(r^2+a^2\cos^2{\theta})\, d\theta^2 \\ \,\,\,+ \left( r^2+a^2+ \frac{2Ma^2 r \sin^2{\theta}}{r^2+a^2\cos^2{\theta}}\right) \sin^2{\theta}\, d\phi^2 - \frac{2Ma r \sin^2{\theta}}{r^2+a^2\cos^2{\theta}} dt\, d\phi $$
The scalar wave equation given below
$$ \frac{1}{\sqrt{-g}}\partial_\mu\left(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\Psi\right) = 0$$
admits a separable solution of the form: $\Psi = e^{-i\omega t} e^{i m \phi} R(r) S(\theta)$.
Is there any other coordinate system that the above wave equation admits separable solution ? or BL coordinates is the only one that we know of ? I would appreciate relevant references.
