  The goal of this paper is to develop a Heine-Stieltjes theory for univariate
linear differential operators of higher order. Namely, for a given given
operator T=\sum_i Q_i(z)d^i/dz^i with polynomial coefficients Q_i(z) set
r=max_i (deg Q_i(z)-i). Following the classical approach of Heine and Stieltjes
we study the multiparameter spectral problem of finding all polynomial V(z) of
degree at most r such that the equation:
  T(z)S(z)+V(z)S(z=0 has for a given positive integer n a polynomial solution
S(z) of degree n. We show that under some mild non-degeneracy assumptions there
exist exactly ((n+r) choose n) such polynomials V_n,i(z). We generalize a
number of classically known results in this area and discuss occurring
degeneracies.
