Local Description of Homogeneous Real Hypersurfaces of the Two-Dimensional Complex Space in Terms of Their Normal Equations

In the paper, a classification of real hypersurfaces of the space $\mathbb{C}^2$ that admit transitive actions of local Lie groups of holomorphic transformations is constructed.
Any nonspherical Levi nondegenerate homogeneous surface is determined by the triple of real coefficients $N^2_{520}$, $N_{440}$, $\operatorname{Im}N_{421}$ of a Moser normal equation. All such surfaces are described by several quadratic curves in the space of above coefficients.