  We provide the first known family of examples of integrable homogeneous
sub-Riemannian structures admitting strictly abnormal geodesics. These examples
were obtained through the analysis of the equivalence problem for
sub-Riemannian Engel structures. We formulate a criterion of strict abnormality
in terms of structure functions of a canonical frame on a sub-Riemannian Engel
manifold as well as estimates on conjugate times.
