  Classification theorems for linear differential equations in two real
variables, possessing eigenfunctions in the form of the polynomials (the
generalized Bochner problem) are given. The main result is based on the
consideration of the eigenvalue problem for a polynomial elements of the
universal enveloping algebras of the algebras
  $sl_3({\bf R})$, $sl_2({\bf R})\oplus sl_2({\bf R})$
  and
  $gl_2 ({\bf R})\ \triangleright\!\!\!< {\bf R}^{r+1}\ , r>0$
  taken in the "projectivized" representations (in differential operators of
the first order in two real variables) possessing an invariant subspace.
General insight to the problem of a description of linear differential
operators possessing an invariant sub-space with a basis in polynomials is
presented. Connection to the recently-discovered quasi-exactly-solvable
problems is discussed.
