  A contact twisted cubic structure (M,C,S) is a 5-dimensional manifold M
together with a contact distribution C and a bundle S of twisted cubics that is
compatible with the conformal symplectic form on C. In Engel's classical work,
the Lie algebra of the exceptional Lie group G_2 was realized as the symmetry
algebra of the most symmetric contact twisted cubic structure; we thus refer to
this one as the contact Engel structure. In the present paper we equip the
contact Engel structure with a smooth section s: M-> S that `marks' a point in
each twisted cubic. We study the local geometry of the resulting structures
(M,C,S,s), which we call marked contact Engel structures. Equivalently, our
study can be viewed as a study of foliations of M by curves whose tangent
directions are everywhere contained in S. We provide a complete set of local
invariants of marked contact Engel structures, we classify all homogeneous
models with symmetry groups of dimension greater than 5, and we prove an
analogue of the classical Kerr theorem from relativity.
