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The goal of this paper was to develop a Heine–Stieltjes theory for univariate linear differential operators of higher order. For a given linear ordinary differential operator      (  z  )  =    ∑   i  =  1   kQ  i   (  z  )  (   d  i   /  d   z  i   )    d    with polynomial coefficients, set  r =max   i =1, …,  k   (deg  Q   i  ( z )− i ). If ( z ) satisfies conditions (i)  r ⩾0 and (ii) deg  Q   k  ( z )= k + r , then we call it a  non‐degenerate higher Lamé operator . Following the classical approach of Heine [ Handbuch der Kugelfunctionen  1 (G. Reimer, Berlin, 1878) 472–479] and Stieltjes [‘Sur certains polynômes qui vérifient une équation différentielle linéaire du second ordre et sur la theorie des fonctions de Lamé’,  Acta Math.  6 (1885) 321–326 (French)], we study the multiparameter spectral problem of finding all polynomials  V ( z ) of degree at most  r  such that for a given positive integer  n , the equation  (  z  )  S  (  z  )  +  V  (  z  )  S  (  z  )  =  0has a polynomial solution  S ( z ) of degree  n . We show that under some mild non‐degeneracy assumptions, there exist exactly(             n    n  +  r)such polynomials  V   n, i  ( z ) whose corresponding eigenpolynomials  S   n, i  ( z ) are of degree  n . We generalize a number of well‐known results in this area and discuss occurring degeneracies.

Algebro‐geometric aspects of Heine–Stieltjes theory