The problem of describing holomorphically homogeneous real hypersurfaces in $\Bbb C^3$

The report outlines approaches and a complete solution to the problem of describing (locally) holomorphically homogeneous hypersurfaces of a 3-dimensional complex space.
The study of the problem was started (20 years ago) in the framework of the coefficient approach using holomorphic normal forms and holomorphic invariants of the varieties under consideration. The bulk of the results, associated with the description of individual classes of homogeneous manifolds (spherical, tubular, Levi-degenerate, surfaces with rich symmetry algebras) was obtained on the basis of methods and classification statements from the theory of Lie algebras. At the final stage of the study, in connection with the possible holomorphic equivalence of the obtained manifolds, the main role is played again by the technique of normal forms and computations (by means of computer algebra) of holomorphic invariants of such varieties.