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

In the homogeneous manifold $\mathbb{E}(-1,\tau ),$ for $-\frac{1}{2}<H<\frac{1}{2},$ {we define a new product compactification in which the slices ${\{t=c\}}_{c\in \mathbb{R}}$ 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 $\mathrm{\Gamma }$ projecting bijectively onto \partial\H^2 we prove the existence of a unique entire $H$-graph that is asymptotic to $\mathrm{\Gamma }$.} We also find necessary and sufficient conditions for the existence of $H$-graphs over unbounded domains having prescribed, possibly infinite boundary data.