  For $0\leq H< 1/2$, 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. When $H=0$ the examples are minimal graphs constructed
by P. Collin and the second author [2].
