On orbits of action of 5-dimensional non-solvable Lie algebras in three-dimensional complex space

After the description by E. Cartan in 1932 holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, a similar study in the 3-dimensional case remains incomplete. In a series of works performed by several international teams of authors, the problem is reduced to describing homogeneous surfaces that are non-degenerate in Levi sense and have exactly 5-dimensional Lie algebras of holomorphic vector fields. In this paper, precisely such homogeneous surfaces are investigated. At the same time, a significant part of the extensive list of abstract 5-dimensional Lie algebras does not provide new examples of homogeneity. A complete description of the orbits of 5-dimensional non-solvable Lie algebras in a three-dimensional complex space, given in the paper, includes examples of new homogeneous hypersurfaces. The presented results bring to finish a large-scale scientific study of interest to various branches of mathematics.