  We study the equivalence problem for $4$-dimensional CR-manifolds of
CR-dimension $1$ and codimension $2$ which are referred to as Engel
CR-manifolds. We construct a canonical Cartan connection on such CR-manifolds
through Cartan equivalence's method. In particular, we give the explicit
expression of $4$ biholomorphic invariants, the annulation of which is a
necessary and sufficient condition for an Engel manifold to be locally
biholomorphic to Beloshapka's cubic in $\mathbb{C}^3$.
