  If $\xi$ is a Killing vector field of the hyperbolic space $\h^3$ whose flow
are parabolic isometries, a surface $\Sigma\subset\h^3$ is a $\xi$-translator
if its mean curvature $H$ satisfies $H=\langle N,\xi\rangle$, where $N$ is the
unit normal of $\Sigma$. We classify all $\xi$-translators invariant by a
one-parameter group of rotations of $\h^3$, exhibiting the existence of a new
family of grim reapers. We use these grim reapers to prove the non-existence of
closed $\xi$-translators.
