Equivariant completions of homogenous algebraic varieties by homogenous divisors

Complete smooth complex algebraic varieties with an almost transitive action of a linear algebraic group are studied. They are classified in the case, when the complement of the open orbit is a homogeneous hypersurface. If the group and the isotropy subgroup at a generic point are both reductive, then there exists a natural one-to-one correspondence between these two-orbit varieties and compact riemannian symmetric spaces of rank one.