It is well-known that an ellipsoid $G$ has the following property (E) and (E* resp.): for any polynomial $f$ (entire function resp.), the solution of the Dirichlet problem for the restriction of $f $ to the boundary of the domain $G$, has an polynomial (entire resp.) extension. In this talk we give a positive answer to the following question of D. Khavinson and H.S. Shapiro for a large class of domains: the ellipsoids are the only domains satsifying (E) or (E*) in this class. The results follow from a more general theorem in the context of so-called Fischer pairs. Another interesting consequence of our methods is the following result: if a polyharmonic entire function of order $ k$ vanishes on $k$ distinct ellipsoids then it vanishes everywhere. This answers a question of W.K. Hayman.