  It is proved that the holomorphic quadratic differential associated to CMC
surfaces in Riemannian products $\mathbb{S}^2\times\Rr$ and $\mathbb{H}^2\times
\Rr$ discovered by U. Abresch and H. Rosenberg could be obtained as a linear
combination of usual Hopf differentials. Using this fact, we are able to extend
it for Lorentzian products. Families of examples of helicoidal CMC surfaces on
these spaces are explicitly described. We also present some characterizations
of CMC rotationally invariant discs and spheres. Finally, after establish some
height and area estimates, we prove the existence of constant mean curvature
Killing graphs.
