On holomorphically homogeneous hypersurfaces in $ \mathbb C^4 $

In connection with the description problem of (locally) holomorphically homogeneous real hypersurfaces of multidimensional complex spaces the case of the space $ \mathbb C^4 $ is discussed (the 2-dimensional case was described by E. Cartan, the study of the 3-dimensional one was completed last year by the author of this report).
Three approaches to the problem are considered:
1) representations of abstract Lie algebras in the form of holomorphic vector fields algebras on the homogeneous hypersurfaces under consideration;
2) a coefficient approach using the properties of normal (canonical) equations of homogeneous manifolds;
3) computer calculations associated with the homogeneity property.
Within the framework of the first approach, in the space $ \mathbb C^4 $, all the Levi nondegenerate orbits of 7-dimensional nilpotent Lie algebras are studied. It is proven that for the entire collection of 180 types of such algebras there are only 4 (up to holomorphic equivalence) nondegenerate surfaces: two quadrics and two generalizations of the known Winkelmann surface from $ \mathbb C^3 $.
For analytic strictly pseudoconvex hypersurfaces of the space $ \mathbb C^4 $, the properties of fourth degree polynomials from the normal Moser equations are studied. It is expected that, by analogy with the 3-dimensional case, the using of these properties will provide a description of large families homogeneous surfaces with rich symmetry algebras.
On the base of algorithms of symbolic mathematics, a large number of examples of affine homogeneous hypersurfaces of the space $ \mathbb C^4 $ (both nondegenerate and degenerate) are constructed.
This work was supported by the Russian Foundation for Basic Research (project No 20-01-00497-a).