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.