Cone structures and path geometries with constant torsion

We study three-dimensional path geometries with nontrivial torsion of maximal rank. We introduce the notion of constant torsion and show that such path geometries are in one-to-one correspondence with certain cone structures modeled on homogeneous ruled surfaces. We also describe these cone structures in terms of solutions to new integrable systems admitting dispersionless, non-isospectral Lax pairs.