  A holomorphic Engel structure determines a flag of distributions
$\mathcal{W}\subset \mathcal{D}\subset \mathcal{E}$. We construct examples of
Engel structures on $\mathbf{C}^4$ such that each of these distributions is
hyperbolic in the sense that it has no tangent copies of $\mathbf{C}$. We also
construct two infinite families of pairwise non-isomorphic Engel structures on
$\mathbf{C}^4$ by controlling the curves $f:\mathbf{C}\to \mathbf{C}^4$ tangent
to $\mathcal{W}$. The first is characterised by the topology of the set of
points in $\mathbf{C}^4$ admitting $\mathcal{W}$-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 $\mathbf{C}^4$.

Complex Variables;Differential Geometry