Consider the differential operator $$ \mathcal{L} = -A \partial_x f_1(x,y) - B \partial_y f_2(x,y) + \frac{A}{2} \partial_x^2 f_1(x,y) + \frac{B}{2} \partial_y^2 f_2(x,y)$$ where $f_1 = (x+1)(M-x-y)$, $f_2 = (y+1)(M-x-y)$, with positive constants $M$, $A$ and $B$. What are the eigenfunctions $u(x,y)$ satisfying $\mathcal{L}u = E u$ with eigenvalue $E$?
I know that the solution should be symmetric under exchange of $x \leftrightarrow y$ and $A \leftrightarrow B$, but I am unable to proceed beyond that.