Geodesic
Local Description of Homogeneous Real Hypersurfaces of the Two-Dimensional Complex Space in Terms of Their Normal Equations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 2, pp. 33-42

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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. 
@article{FAA_2000_34_2_a3,
     author = {A. V. Loboda},
     title = {Local {Description} of {Homogeneous} {Real} {Hypersurfaces} of the {Two-Dimensional} {Complex} {Space} in {Terms} of {Their} {Normal} {Equations}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {33--42},
     publisher = {mathdoc},
     volume = {34},
     number = {2},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2000_34_2_a3/}
}
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A. V. Loboda. Local Description of Homogeneous Real Hypersurfaces of the Two-Dimensional Complex Space in Terms of Their Normal Equations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 2, pp. 33-42. http://geodesic.mathdoc.fr/item/FAA_2000_34_2_a3/