  We study constant mean curvature 1/2 surfaces in H2xR that admit a
compactification of the mean curvature operator. We show that a particular
family of complete entire graphs over H2 admits a structure of infinite
dimensional manifold with local control on the behaviors at infinity. These
graphs also appear to have a half-space property and we deduce a uniqueness
result at infinity. Deforming non degenerate constant mean curvature 1/2
annuli, we provide a large class of (non rotational) examples and construct
(possibly embedded) annuli without axis, i.e. with two vertical, asymptotically
rotational, non aligned ends.
