On the coefficients description for affine-homogeneous real hypersurfaces of the three-dimensional complex space

The classification problem for holomorphically homogeneous real hypersurfaces of complex 3-dimensional spaces includes a simpler question concerning affine homogeneity. Using the technique of matrix Lie algebras, a complete description is obtained for the class of affinely homogeneous hypersurfaces of tubular type, characterized by the natural constraints on the coefficients of the 2nd order of their canonical (affine) equations. Due to the computer solving of a large system of quadratic equations, one can construct a complete list of matrix Lie algebras corresponding to the desired hypersurfaces. Then, these algebras are integrated. Many of the constructed hypersurfaces are reduced by holomorphic transformations to the known homogeneous manifolds, some other examples are new, even in the sense of the holomorphic homogeneity. Using the proposed scheme, it is possible to obtain a complete classification of affine-homogeneous real hypersurfaces of the complex 3-dimensional space.