On degeneracy of orbits of nilpotent Lie algebras

In the paper we discuss $7$-dimensional orbits in $\mathbb{C}^4$ of two families of nilpotent $7$-dimensional Lie algebras; this is motivated by the problem on describing holomorphically homogeneous real hypersurfaces. Similar to nilpotent $5$-dimensional algebras of holomorphic vector fields in $\mathbb{C}^3$, the most part of algebras considered in the paper has no Levi non-degenerate orbits. In particular, we prove the absence of such orbits for a family of decomposable $7$-dimensional nilpotent Lie algebra ($31$ algebra). At the same time, in the family of $12$ non-decomposable $7$-dimensional nilpotent Lie algebras, each containing at least three Abelian $4$-dimensional ideals, four algebras has non-degenerate orbits. The hypersurfaces of two of these algebras are equivalent to quadrics, while non-spherical non-degenerate orbits of other two algebras are holomorphically non-equivalent generalization for the case of $4$-dimensional complex space of a known Winkelmann surface in the space $\mathbb{C}^3$. All orbits of the algebras in the second family admit tubular realizations.