On integrability of certain rank 2 sub-Riemannian structures

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.

In the second version we restructured the material, improved non-existence result in dimension 7 (to degree 6 using the modular approach), and updated references. We also refined the algorithm, and we attach the corresponding commented Maple file (together with PDF outputs of its work for several cases) as the supplement

References in corpus (5)

• The non-integrability of the Zipoy-Voorhees metric
• The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta
• Reducibility of Valence-3 Killing Tensors in Weyl's Class of Stationary and Axially Symmetric Space-Times
• Sub-Riemannian geodesics on the free Carnot group with the growth vector (2,3,5,8)
• Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups

Cited by in corpus (3)

• Killing Tensors in Koutras-McIntosh Spacetimes
• Killing tensors in stationary and axially symmetric space-times
• Chaotic subRiemannian geodesic flow in $J^2({\mathbb{R}}^2,\mathbb{R})$