Decomposable five-dimensional Lie algebras in the problem of holomorphic homogeneity in $\mathbb{C}^3$

In connection with the problem of describing holomorphically homogeneous real hypersurfaces in the space $\mathbb{C}^3$, we study five-dimensional real Lie algebras realized as algebras of holomorphic vector fields on such manifolds. We prove that if on a holomorphically homogeneous real hypersurface $M$ of the space $\mathbb{C}^3$, there is a decomposable, solvable, five-dimensional Lie algebra of holomorphic vector fields having a full rank near some point $P\in M$, then this surface is either degenerate near $P$ in the sense of Levy or is a holomorphic image of an affine-homogeneous surface.