Homogeneous hypersurfaces in $\text{CC}3$, associated with a model CR-cubic

The model 4-dimensional CR-cubic in $\text{CC}3$ has the following “model” property: it is (essentially) the unique locally homogeneous 4-dimensional CR-manifold in $\text{CC}3$ with finite-dimensional infinitesimal automorphism algebra $\mathfrak{g}$ and non-trivial isotropy subalgebra. We study and classify, up to local biholomorphic equivalence, all $\mathfrak{g}$-homogeneous hypersurfaces in $\text{CC}3$ and also classify the corresponding local transitive actions of the model algebra $\mathfrak{g}$ on hypersurfaces in $\text{CC}3$.