Invariant $$\lambda $$-translators in Lorentz-minkowski space

Abstract 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.