I am looking for the solution of the Helmoltz equation (or even Laplace, if not available) on the torus, that is, the manifold of line element
\begin{equation} ds^2 = (c + a \sin(\theta))^2 d\varphi^2 + a^2 d\theta^2 \end{equation}
I can find papers for the flat torus and the Helmoltz equation in toroidal coordinates, but they do not seem to have any direct link with the equation, which is
\begin{equation} \frac{1}{a^2(c+a \sin(\theta))}\partial_\theta ( (c+a \sin(\theta))\partial_\theta f) + \frac{1}{(c+a \sin(\theta))^2} \partial^2_\varphi f = k \end{equation}
Is there any coordinate transform that will turn this into the Helmoltz equation for toroidal coordinates (at fixed $\eta$), or is the solution something else entirely?