Affine homogeneous strictly pseudoconvex hypersurfaces of the type   (1/2,0) in $\mathbb{C}^3$
  In this paper we denote a type of affine homogeneous real hypersurface of $\mathbb{C}^3$ and present a classification of homogeneous surfaces of the type (1/2,0). The result was obtained by reducing the classification problem mentioned above to the problem of solving a system of nonlinear (quadratic) equations. Solutions of such system represent all the Lie algebras corresponding to the homogeneous surfaces under consideration. Some details of integration procedure are discussed for obtained algebras. 
A.V. Atanov, A.V. Loboda, A.V. Shipovskaya
math.CV