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.