The completeness problem on the pseudo-homothetic Lie group

Let us call pseudo-homothetic group the non-unimodular 3-dimensional Lie group that is the semi-direct product of $\mathbb{R}$ acting non-semisimply on $\mathbb{R}^2$. In this article, we solve the geodesic completeness problem on this Lie group. In particular, we exhibit a family of complete metrics such that all geodesics have bounded velocity. As an application, we show that the set of complete metrics is not closed.