We study CR quadrics satisfying a symmetry property $(tilde S)$ which is slightly weaker than the symmetry property $(S)$, recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of i
nfinitesimal automorphisms of the quadric. We characterize quadrics satisfying the $(tilde S)$ property in terms of their Levi-Tanaka algebras. In many cases the $(tilde S)$ property implies the $(S)$ property; this holds in particular for compact quadrics. We also give a new example of a quadric such that the dimension of the algebra of positive-degree infinitesimal automorphisms is larger than the dimension of the quadric.
We investigate the $CR$ geometry of the orbits $M$ of a real form $G_0$ of a complex simple group $G$ in a complex flag manifold $X=G/Q$. We are mainly concerned with finite type, Levi non-degeneracy conditions, canonical $G_0$-equivariant and Mostow
fibrations, and topological properties of the orbits.
We compute the Euler-Poincare characteristic of the homogeneous compact manifolds that can be described as minimal orbits for the action of a real form in a complex flag manifold.
