Lagrangian Engel Structures

We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures. A \em{Lagrangian} Engel structure is an Engel 2-plane field on a symplectic 4-manifold for which the 2-planes are Lagrangian with respect to the symplectic structure. We solve the equivalence problems for Lagrangian Engel structures and use the resulting structure equations to classify homogeneous Lagrangian Engel structures. This allows us to determine all compact, homogeneous examples. Compact manifolds that support homogeneous Lagrangian Engel structures are diffeomorphic to quotients of one of a determined list of nilpotent or solvable 4-dimensional Lie groups by co-compact lattices.