Sub-Riemannian Engel sphere

The structure of the intersection of the sub-Riemannian sphere on the Engel group with a two-dimensional invariant set of discrete symmetries is described: regularity, analytic properties, exp-log category, Whitney stratification, multiplicity of points, characterization in terms of abnormal trajectories, conjugate points and Maxwell points, and explicit expressions for the sub-Riemannian distance to singular points.