Homogeneous Surfaces in $$\mathbb C^4$$ Associated with a 5-Dimensional Completely Nondegenerate Cubic Model Surface of CR-Type $$(1,3)$$

Abstract

The action on the space $$\mathbb C^4$$ of the 7-dimensional Lie group of infinitesimal holomorphic automorphisms of a completely nondegenerate cubic model surface $$Q$$ of CR-type $$(1,3)$$ is considered. All orbits of the given action are found and their biholomorphic classification is given. One of the orbits coincides with the surface $$Q$$ (the 5-dimensional orbit), two orbits are 6-dimensional, and the remaining part of the space $$\mathbb C^4$$ , the complement to the above orbits, foliates into 7-dimensional real orbits. It is also proved that the algebra of infinitesimal holomorphic automorphisms of all orbits, except for the twelve holomorphically degenerate orbits, coincides with the algebra of the surface $$Q$$ .