Maximally homogeneous para-CR manifolds of semisimple type

An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together with a field $K \in \Gamma({\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras.