Orthogonal polynomials and diffusion operators

We study the following problem: describe triplets (?,g,?) where g=(g ij (x)) is (co)metric associated with symmetric second order differential operator L(f)=1 ?? ? i (g ?? j f) defined on a domain ? of ? d (that L diffusion reversible measure ?(dx)=?(x)dx) and such that there exists an orthonormal basis ? 2 (?) made polynomials which are at same time eigenvectors L, ranked according to their natural degree. reduce this problem certain algebraic (for any d) we find all solutions for d=2 when compact. Namely, in dimension d=2, up affine transformations, 10 compact domains plus one-parameter family. The proof list exhaustive relies Plücker-like formulas projective dual curves applied complexification ??. then some geometric origins these various models. also give description non-compact cases dimension.