Constant mean curvature graphs with prescribed asymptotic values in $\mathbb{E}(-1,\tau)$

In the homogeneous manifold $\mathbb{E}(-1,\tau),$ for $-\tfrac{1}{2}<H<\tfrac{1}{2},$ {we define a new product compactification in which the slices $\left\{t=c\right\}_{c\in\R}$ are rotational $H$-surfaces. This product compatification is the natural setting where it makes sense to study the asymptotic Dirichlet Problem for the constant mean curvature equation. Indeed, for every rectifiable curve $\Gamma$ projecting bijectively onto \partial\H^2 we prove the existence of a unique entire $H$-graph that is asymptotic to $\Gamma$.} We also find necessary and sufficient conditions for the existence of $H$-graphs over unbounded domains having prescribed, possibly infinite boundary data.