  It has been recently shown by Abresch and Rosenberg that a certain Hopf
differential is holomorphic on every constant mean curvature surface in a
Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this
paper we describe all the surfaces with holomorphic Hopf differential in the
homogeneous 3-manifolds isometric to H^2xR or having isometry group isomorphic
either to the one of the universal cover of PSL(2,R), or to the one of a
certain class of Berger spheres. It turns out that, except for the case of
these Berger spheres, there exist some exceptional surfaces with holomorphic
Hopf differential and non-constant mean curvature.
