Homogeneous $$\mathrm{CR}$$-Manifolds in $$\mathbb{C}^4$$

Homogeneous model submanifolds of $$\mathrm{CR}$$ -type $$(1, 3)$$ in the complex space $$\mathbb C^4$$ are investigated. In the course of the study, the moduli space of five-dimensional model surfaces of the Bloom–Graham type $$((2, 1),(3, 1),(4, 1))$$ are found. It is also shown that, among the model surfaces of this type, exactly one surface has the property of holomorphic homogeneity, which is equivalent to the tubular surface $$\mathcal {C}$$ over an affinely homogeneous curve in $$\mathbb R ^4$$ . The paper describes a family of six-dimensional holomorphically homogeneous surfaces obtained as orbits of the action of the group of holomorphic automorphisms of $$\mathcal C$$ and classifies them from the point of view of the corresponding model surfaces.