Classification of invariant Gauss curvature solitons in the Heisenberg space $\nil$

In this paper, we classify all solitons of the Gauss curvature flow in the three-dimensional Heisenberg group $\mathrm{Nil}_3$ that are
invariant under a one-parameter group of ambient isometries. By means of the four canonical types of Killing vector
fields and the three families of invariant surfaces (vertical translations, horizontal translations,
and helicoidal motions), we analyze the twelve resulting types of possible solitons. In some cases, there do not exist any invariant solitons; in others, we find explicit parametrizations, or describe their geometric properties.