  The classical Heun equation has the form {Q(z) d^2/dz^2 +P(z) d/dz
+V(z)}S(z)=0 where Q(z) is a cubic, P(z) at most quadratic and V(z) linear
polynomials resp. In the second half of the 19-th century E.Heine and
T.STieltjes initiated the study of the set of all V(z) such that the above
equation has a polynomial solution S(z) of a given degree n. The main goal of
the present paper is to study the union of the roots of the latter set of
V(z)*s when n->oo. We formulate an intriguing conjecture of K.Takemura
describing the limiting set and give a substantial amount of additional
information.
