Left-invariant Sub-Riemannian Engel structures: abnormal geodesics and integrability

We provide the first known family of examples of integrable sub-Riemannian structures admitting strictly abnormal geodesics. These examples were obtained through the analysis of the equivalence problem for sub-Riemannian Engel structures. We proved that 6 invariants define a sub-Riemannian Engel structure and described the classification of left-invariant sub-Riemannian structures of Engel type. As an application of these results we provide a criterion of strict abnormality of geodesics as well as estimates on conjugate times in terms of the obtained invariants.