Partial differential equations having orthogonal polynomial solutions

We show that if a second order partial differential equation: L[u]:= Auxx + 2Buxy + Cuyy + Dux + Euy = λnu has orthogonal polynomial solutions, then the differential operator L[·] must be symmetrizable and can not be parabolic in any nonempty open subset of the plane. We also find Rodrigues type formula for orthogonal polynomial solutions of such differential equations.