  We consider the nilpotent left-invariant sub-Riemannian structure on the
Engel group. This structure gives a fundamental local approximation of a
generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point
(in particular, of the kinematic models of a car with a trailer). On the other
hand, this is the simplest sub-Riemannian structure of step three. We describe
the global structure of the cut locus (the set of points where geodesics lose
their global optimality), the Maxwell set (the set of points that admit more
than one minimizer), and the intersection of the cut locus with the caustic
(the set of conjugate points along all geodesics). The group of symmetries of
the cut locus is described: it is generated by a one-parameter group of
dilations $\mathbb{R}_+$ and a discrete group of reflections $\mathbb{Z}_2
\times \mathbb{Z}_2 \times \mathbb{Z}_2$. The cut locus admits a stratification
with 6 three-dimensional strata, 12 two-dimensional strata, and 2
one-dimensional strata. Three-dimen-sional strata of the cut locus are Maxwell
strata of multiplicity 2 (for each point there are 2 minimizers).
Two-dimensional strata of the cut locus consist of conjugate points. Finally,
one-dimensional strata are Maxwell strata of infinite multiplicity, they
consist of conjugate points as well. Projections of sub-Riemannian geodesics to
the 2-dimensional plane of the distribution are Euler elasticae. For each point
of the cut locus, we describe the Euler elasticae corresponding to minimizers
coming to this point. Finally, we describe the structure of the optimal
synthesis, i.e., the set of minimizers for each terminal point in the Engel
group.
