Torsion-free path geometries and integrable second order ODE systems

A search for invariants of second order ODE systems under the class of point transformations, which mix the parameter and the dependent variables, uncovers a torsion tensor generalizing part of the curvature tensor of an affine connection. We study the geometry of ODE systems for which this torsion vanishes. These are the ODE systems for which deformations of solutions fixing a point constitute a field of Segré varieties in the tangent bundle of the locally defined space of solutions. Conversely, a field of Segré varieties for which certain differential invariants vanish induces a torsion-free ODE system on the space of solutions to a natural PDE system. The geometry on the solution space is used to produce first integrals for torsion-free ODE systems, given as algebraic invariants of a curvature tensor involving up to fourth derivatives of the equations. In the generic case, there are enough first integrals to solve the equations explicitly in spite of the absence of symmetry. In the case of torsion-free ODE pairs, the field of Segré varieties is equivalent to a half-flat split signature conformal structure, and we characterize in terms of curvature those systems having an abundance of totally geodesic surfaces.