Paraconformal geometry of n th-order ODEs, and exotic holonomy in dimension four

If n=2 or 3M admits an affine paraconformal connection with no torsion. In these cases additional constraints can be imposed on the ODE so that M admits a projective structure if n=2, or an Einstein-Weyl structure if n=3. The third-order ODE can in this case be reconstructed from the Einstein-Weyl data.