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 $0<H\le \frac{1}{2}$, genus one and $k\ge 2$ ends ($k=2$ only for one of these families). These ends are asymptotic to vertical $H$-cylinders for $0<H<\frac{1}{2}$. 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.