Paraconformal structures, ordinary differential equations and totally geodesic manifolds

We construct point invariants of ordinary differential equations of arbitrary order that generalise the Tresse and Cartan invariants of equations of order two and three, respectively. The vanishing of the invariants is equivalent to the existence of a totally geodesic paraconformal structure which consists of a paraconformal structure, an adapted GL(2,R)GL(2,R)-connection and a two-parameter family of totally geodesic hypersurfaces on the solution space. The structures coincide with the projective structures in dimension 2 and with the Einstein–Weyl structures of Lorentzian signature in dimension 3. We show that the totally geodesic paraconformal structures in higher dimensions can be described by a natural analogue of the Hitchin twistor construction. We present a general example of Veronese webs that generalise the hyper-CR Einstein–Weyl structures in dimension 3. The Veronese webs are described by a hierarchy of integrable systems.