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

Canonical Cartan connection for $4$-dimensional CR-manifolds belonging to general class ${\sf II}$