I'll introduce a class of curves called Lewy curves in para-CR
geometry, following H.Lewy's original definition in CR geometry. I'll
show that in dimension 3 the curves are always solutions to a 2nd order
system of ODEs, meaning geometrically, they define a path geometry on a
manifold. This path geometry uniquely determines a para-CR structure.
In higher dimensions, the Lewy curves define a path geometry if and only
if the para-CR structure is flat. In general they are described by a
system of ODEs of higher order.
Finally, I'll present a characterization of path geometries of Lewy
curves in the class of general path geometries. I'll also discuss
relations between the Lewy curves and chains - another class of curves
canonically associated to para-CR and CR structures.
The talk is based on a joint work with O.Makhmali.