Invariant $\lambda$-translators in $\mathbb{S}^2\times\mathbb{R}$

A $\lambda$-translator in $\mathbb{S}^2\times\mathbb{R}$ is an oriented surface whose mean curvature $H$ satisfies $H=\langle N,\partial_z\rangle+\lambda$, where $N$ is the unit normal, $\partial_z$ is the vertical Killing vector field tangent to the fibers of the submersion and $\lambda\in\mathbb{R}$.

When $\lambda=0$ we fall into the class of translators.

In this paper, we classify all $\lambda$-translators that are invariant by a one-parameter group of rotations and by vertical translations of $\mathbb{S}^2\times\mathbb{R}$.