  We investigate the strong asymptotics of Heine-Stieltjes polynomials -
polynomial solutions of a second order differential equations with complex
polynomial coefficients. The solution is given in terms of critical measures
(saddle points of the weighted logarithmic energy on the plane), that are
tightly related to quadratic differentials with closed trajectories on the
plane. The paper is a continuation of the research initiated in
[arXiv:0902.0193]. However, the starting point here is the WKB method, which
allows to obtain the strong asymptotics.
