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We provide the first known family of examples of integrable homogeneoussub-Riemannian structures admitting strictly abnormal geodesics. These exampleswere obtained through the analysis of the equivalence problem forsub-Riemannian Engel structures. We formulate a criterion of strict abnormalityin terms of structure functions of a canonical frame on a sub-Riemannian Engelmanifold as well as estimates on conjugate times.

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