  We construct point invariants of ordinary differential equations that
generalise the Cartan invariants of equations of order two and three. The
vanishing of the invariants is equivalent to the existence of a totally
geodesic paraconformal structure which consist of a paraconformal structure, an
adapted $GL(2,\R)$-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 which correspond to the hyper-CR Einstein-Weyl structures in
dimension 3. The Veronese webs are described by a hierarchy of integrable
systems.
