  Given $\lambda\in\mathbb{R}$ and $\textbf{v}\in\mathbb{L}^3$, a
$\lambda$-translator with velocity $\textbf{v}$ is an immersed surface in
$\mathbb{L}^3$ whose mean curvature satisfies $H=\langle
N,\textbf{v}\rangle+\lambda$, where $N$ is a unit normal vector field. When
$\lambda=0$, we fall into the class of translating solitons of the mean
curvature flow. In this paper we study $\lambda$-translators in $\mathbb{L}^3$
that are invariant under a 1-parameter group of translations and rotations. The
former are cylindrical surfaces and explicit parametrizations are found,
distinguishing on the causality of both the ruling direction and the
$\lambda$-translators. In the case of rotational $\lambda$-translators we
distinguish between spacelike and timelike rotations and exhibit the
qualitative properties of rotational $\lambda$-translators by analyzing the
non-linear autonomous system fulfilled by the coordinate functions of the
generating curves.
