We study regular deformations of constant mean curvature (CMC) $1/ 2$ surfaces with vertical ends in $\mathbb{H}^2 \times \mathbb{R}$ using a suitable extension of the mean curvature operator at infinity. The two main results are the following :

• We show that the moduli space of CMC-$1/ 2$ entire graphs with vertical end can endowed with a strucutre of infinite dimensional smooth manifold based on $\mathcal{C}^{2, \alpha} \times \mathbb{R}$
• We can deform CMC-$1/ 2$ rotational annuli so that the ends of the resulting annuli are still asymptotically rotational but no longer with the same axis.