We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded Engel CR manifold.

We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves "almost" like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of "semi-flat" manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection.In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.

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We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.

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