Translators of the mean curvature flow in the special linear group SL(2,R)

Translators in the special linear group SL(2,R) are surfaces whose mean curvature H and unit normal vector N satisfy H= N,X, where X is a fixed Killing vector field. In this paper we study and classify those translators that are invariant by a one-parameter group of isometries. By the Iwasawa decomposition, there are three types of such groups. The dimension of the Killing vector fields is $4$ and an exhaustive discussion is done for each one of the Killing vector fields and each of the invariant surfaces. In some cases, explicit parametrizations of translators are obtained.