Canonical Cartan connection for $5$-dimensional CR-manifolds belonging to general class ${\sf III_2}$

We study the equivalence problem for CR-manifolds belonging to general class III_2, i.e. the 5-dimensional CR-manifolds of CR-dimension 1 and codimension 3 whose CR-bundle satisfies a certain degeneracy condition. For such a CR-manifold M, we construct a canonical Cartan connection on a 6-dimensional principal bundle P on M. This provides a complete set of biholomorphic invariants for M.