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

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