Here I want to solve a second order PDE symmetrically depending on two variables $$ 3(\partial_{\alpha\alpha}f+\partial_{\beta\beta}f-2\partial_{\alpha\beta}f)+2(\cot\alpha)\partial_\alpha f +2(\cot\beta)\partial_\beta f=cf $$ where $c$ is a constant.

Please do you see any way of solving it for all $c$?

I tried the change of variable $$ x=\sin((\alpha-\beta)/2),\, y=\sin((\theta+\delta)/2) $$ and it's transformed into the following PDE $$ 3(1-x^2)\partial_{xx}g-5x\partial_x g+\frac{2(1-y^2)}{y^2-x^2}(y\partial_y g-x\partial_x g)=cg. $$ It now does not have second order term in $y$, but is no longer symmetric. I am not sure how to continue.