  The well-known Heun equation has the form: Q(z)S''(z)+P(z)S'(z)+V(z)S(z)=0
where Q(z) is a cubic complex polynomial, P(z) and V(z) are polynomials of
degrees at most 2 and 1 resp. One of the classical problems about the Heun
equation is for a given positive integer N to find all possible linear
polynomials V(z) such that the latter equation has a polynomial solution S(z)
of degree N. Below we prove a conjecture of the 2nd author claiming that the
union of roots of such V(z)'s for a given N tends when N->oo to a certain
compact connecting the three roots of Q(z) and given by the condition that a
certain natural abelian integral is real-valued.
