Homogeneous nondegenerate hypersurfaces in \(\mathbb{C}^3\) with two-dimensional isotropy groups (Q1812426)

The author announces a classification (up to local holomorphic equivalence) of homogeneous nonspherical real hypersurfaces \(M\subset \mathbb{C}^3\) with a 2-dimensional isotropy group \(\Aut_0(M)\) and positive definite Levi form. The methods are those of \textit{S. S. Chern} and \textit{J. K. Moser} [Acta Math. 133, 219--271 (1974; Zbl 0302.32015)] (i.e., local normal forms for the equations of the hypersurfaces at hand). Only a sketchy indication of what the proof might be is given.