Conformally homogeneous Lorentzian spaces

We prove that if a 1-connected non-conformally flat conformal Lorentzian manifold $(M,c)$ admits a connected essential transitive group of conformal transformations, then there exists a metric $g\in c$ such that $(M,g)$ is a complete homogeneous plane wave. This finishes the classification of 1-connected Lorentzian manifolds, which admit transitive essential conformal group. We also prove that the group of conformal transformations of a non-conformally flat 1-connected homogeneous plane wave $(M,g)$ consists of homotheties, and it is a 1-dimensional extension of the group of isometries.