The formulas for the lower Taylor coefficients of homogeneous surfaces

This article is related to the descriptions problem for holomorphically homogeneous real hypersurfaces in three-dimensional complex spaces. A common coefficients approach is developing to the study of homogeneity on the base of symbolic mathematics and computing.
The aim of this approach is to look for opportunities to describe homogeneous real hypersurfaces of a $3$-dimensional complex space in terms of the Taylor coefficients of their defining functions.
The result of A. V. Loboda (2000) may be called here as a reference point. It allows to describe homogeneous hypersurfaces in two-dimensional complex space in terms of triple of real Taylor coefficients. This triple itself satisfies certain polynomial constraints.
To study the problem in a much more complicated three-dimensional case it is necessary to work with a large amount of information. In this article, it is done by a Maple symbolic computation computer package.
In this article one particular case is considered of strictly pseudo-convex real hypersurfaces in which foreseeable formulas can be obtained for the parameters and coefficients of the problem under consideration.
The main result of this paper is the reduction the number of parameter describing the property of homogeneity from $16$ coefficients (according to the result of the author and A. V. Loboda, 2015) to $7$ ones. The obtained polynomial system of restrictions on these coefficients, involves five equations. Its consequence is the estimate of the number of homogeneous surfaces of studied class.
In the article the relations are given between the parameters of holomorphic vector fields tangent to the studied homogeneous surfaces. In the case under discussion, these relationships also have a relatively simple form.
The article describes an example confirming the validity of the formulas obtained in it by comparing them with coefficients of concrete homogeneous surface.