Genus one $H$-surfaces with $k$-ends in $\mathbb{H}^{2}\times\mathbb{R}$

We construct two different families of properly Alexandrov-immersed surfaces in \mathbb{H}^{2}\times \mathbb{R} with constant mean curvature 0<H\leq 1/2 , genus one and k\geq2 ends ( k=2 only for one of these families). These ends are asymptotic to vertical H -cylinders for 0<H<1/2 . This shows that there is not a Schoen-type theorem for immersed surfaces with positive constant mean curvature in \mathbb{H}^{2}\times\mathbb{R} . These surfaces are obtained by means of a conjugate construction.