Vitushkin's Germ Theorem for Engel-Type CR Manifolds

We study real analytic CR manifolds of CR dimension $1$ and codimension $2$ in the three-dimensional complex space. We prove that the germ of a holomorphic mapping between “nonspherical” manifolds can be extended along any path (this is an analog of Vitushkin's germ theorem). For a cubic model surface (“sphere”), we prove an analog of the Poincaré theorem on the mappings of spheres into $\mathbb~C^2$. We construct an example of a compact “spherical” submanifold in a compact complex $3$-space such that the germ of a mapping of the “sphere” into this submanifold cannot be extended to a certain point of the “sphere.”