On Implicit ODEs with Hexagonal Web of Solutions

Abstract

Solutions of an implicit ODE form a web. Already for a cubic ODE the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus, any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. In this paper the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity assumptions. Geometric meaning of these assumptions is that the surface defined by an implicit ODE in the space of 1-jets is smooth as well as the criminant of the ODE, which is a critical set of surface projection into the plane of solutions.