Invariant $λ$-translators for the Gauss curvature flow in Euclidean space

A $λ$-translator is a surface in Euclidean space $\mathbb{R}^3$ whose Gauss curvature $K$ satisfies $K=\langle N, \vec{v} \rangle +λ$, where $N$ is the Gauss map, $\vec{v}$ is a fixed direction, and $λ\in \mathbb{R}$. In this paper, we classify all $λ$-translators that are invariant by a one-parameter group of translations and a one-parameter group of rotations.