For
$0\le H<\frac{1}{2}$, we construct
entire
$H$graphs
in
${\mathbb{H}}^{2}\times \mathbb{R}$
that are parabolic and not invariant by one parameter groups of isometries of
${\mathbb{H}}^{2}\times \mathbb{R}$. Their asymptotic
boundaries are
$({\partial}_{\infty}{\mathbb{H}}^{2})\times \mathbb{R}$;
they are dense at infinity. Previously, the only known examples of entire
$H$graphs,
$0<H<\frac{1}{2}$,
were conformally hyperbolic invariant surfaces. When
$H=0$,
the examples are minimal graphs constructed by P. Collin and the second
author.
