Cohomogeneity one dynamics on three dimensional Minkowski space

In the paper, we give a classification of closed and connected Lie groups, up to conjugacy in $\operatorname{Iso}({\mathbb{R}^{1,2}})$, acting by cohomogeneity one on the three dimensional Minkowski space $\mathbb{R}^{1,2}$ in both proper and nonproper dynamics. Then we determine causal properties and types of the orbits.