Exotic Holomorphic Engel Structures on C4

A holomorphic Engel structure determines a flag of distributions (Formula presented.). We construct examples of Engel structures on (Formula presented.) such that each of these distributions is hyperbolic in the sense that it has no tangent copies of (Formula presented.). We also construct two infinite families of pairwise non-isomorphic Engel structures on (Formula presented.) by controlling the curves (Formula presented.) tangent to (Formula presented.). The first is characterised by the topology of the set of points in (Formula presented.) admitting (Formula presented.)-lines and the second by a finer geometric property of this set. A consequence of the second construction is the existence of uncountably many non-isomorphic holomorphic Engel structures on (Formula presented.).