Vertically invariant minimal surfaces in unimodular semidirect products

A surface in a three-dimensional metric Lie group $G$ is said invariant if it is invariant with respect to a one-dimensional subgroup $\Gamma$ of the isometry group of $G$. Is this work we focus on unimodular metric Lie groups $G$ that can be written as a semidirect product of the form $\mathbb{R}^2\rtimes_A \mathbb{R}$ for certain matrix $A\in \mathcal{M}_2(\mathbb{R})$ and study the minimal surfaces which are invariant under the group $\Gamma$ generated by left translations by elements in the vertical axis $\{0\}\rtimes\mathbb{R}$. We will call these surfaces vertically invariant. In particular, we describe new examples of minimal surfaces in $\widetilde{E}(2)$ which are vertically invariant.