On integrability of certain rank 2 sub-Riemannian structures

TL;DR

This paper investigates the integrability of specific rank 2 sub-Riemannian structures on low-dimensional manifolds, revealing that many lack polynomial integrals beyond obvious symmetries, suggesting non-integrability of their geodesic flows.

Contribution

It provides new results on the non-integrability of certain rank 2 sub-Riemannian structures in dimensions 6, 7, and 8, highlighting the absence of low-degree polynomial integrals.

Findings

Many structures have no polynomial integrals beyond Killing fields and the Hamiltonian.

The geodesic flows of these structures are generally non-integrable.

Symmetry groups are large but do not imply integrability.

Abstract

We discuss the integrability of rank 2 sub-Riemannian structures on low-dimensional manifolds, and then prove that some structures of that type in dimension 6, 7 and 8 have a lot of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing fields and the Hamiltonian, thus indicating non-integrability of the corresponding geodesic flows.