  An Engel structure is a maximally non-integrable field of two-planes tangent
to a four-manifold. Any two such structures are locally diffeomorphic. We
investigate the space of global deformations of canonical Engel structures
arising out of contact three-manifolds. The main tool is Cartan's method of
prolongation and deprolongation which lets us pass back and forth between
certain Engel four-manifolds and contact three-manifolds. Every Engel manifold
inherits a natural one-dimensional foliation. Its leaves are the fibers of the
map from Engel to contact manifold, when this map exists. The foliation has a
transverse contact structure and tangential real projective structure. As an
application of our investigations, we show that a canonical Engel structure on
real projective three-space times an interval corresponds to geodesic flow on
the two-sphere, and that a subspace of its Engel deformations corresponds to
the space of Zoll metrics on the two-sphere.
