I wish to solve a given PDE with ellipsoidal symmetry (I'm not sure the wording is correct). For example it could be a PDE describing the temperature field in a bottle. I am thinking of using the symmetry to solve my problem: I should be able to define a mapping from the ellipse to a sphere and obtain a new PDE that has spherical symmetry and is easily solvable. It seems reasonable to me although I can't proof it will give me the correct solution. The idea is very simple but I can't find any papers discussing "solving PDE with ellipsoidal symmetry."
Can it work? Any references discussing this?