Cohomogeneity one actions on anti de Sitter spacetimes

In this paper we classify, up to orbit equivalence, cohomogeneity one actions of connected closed Lie subgroups of $U(1,n)$ on the $(2n+1)$-dimensional anti de Sitter spacetime $Ad{S}^{2n+1}$. We also give some new examples of nonproper cohomogeneity one actions on $Ad{S}^{n+1}$ and determine parabolic Lie subgroups of $SO(2,n)$ and their orbits in $Ad{S}^{n+1}$