Eth-th/92-21 Cpt-92/p.2708

Lie-algebraic approach to the theory of polynomial solutions. II. Differential equations in one real and one Grassmann variables and 2x2 matrix differential equations ABSTRACT A classification theorem for linear differential equations in two variables (one real and one Grassmann) having polynomial solu-tions(the generalized Bochner problem) is given. The main result is based on the consideration of the eigenvalue problem for a polynomial element of the universal enveloping algebra of the algebra osp(2, 2) in the " projectivized " representation (in differential operators of the first order) possessing an invariant subspace. A classification of 2 x 2 matrix differential equations in one real variable possessing polynomial solutions is described. Connection to the recently-discovered quasi-exactly-solvable problems is discussed.