A contact twisted cubic structure({mathcal M},mathcal {C},{varvec{upgamma }}) is a 5-dimensional manifold {mathcal M} together with a contact distribution mathcal {C} and a bundle of twisted cubics {varvec{upgamma }}subset mathbb {P}(mathcal {C}) compatible with the conformal symplectic form on mathcal {C}. The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group mathrm {G}_2. In the present paper we equip the contact Engel structure with a smooth section sigma : {mathcal M}rightarrow {varvec{upgamma }}, which “marks” a point in each fibre {varvec{upgamma }}_x. We study the local geometry of the resulting structures ({mathcal M},mathcal {C},{varvec{upgamma }}, sigma ), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of {mathcal M} by curves whose tangent directions are everywhere contained in {varvec{upgamma }}. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension ge 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.