Invariant $\lambda$-translators in the Heisenberg group

We study oriented surfaces in the Heisenberg space $\mathrm{Nil}_3$ whose mean curvature $H$ at each point is $H=\langle N,\partial_z\rangle+\lambda$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field and $\lambda\in\mathbb{R}$.

These surfaces are known as $\lambda$-translators and generalize, among others, minimal and positive constant mean curvature surfaces, and also translating solitons of the mean curvature flow.

The objective in this paper is to classify $\lambda$-translators invariant by the following one-parameter groups of isometries of $\mathrm{Nil}_3$: left-translations, rotations and helicoidal motions.