Integrability of Homogeneous Exact Magnetic Flows on Spheres

Abstract

We consider motion of a material point placed in a constant homogeneous magnetic field in and also motion restricted to the sphere.While there is an obvious integrability of the magnetic system in, the integrability of the system restricted to the sphere is highly nontrivial. We provecomplete integrability of the obtained restricted magnetic systems for. The first integrals of motion of the magnetic flows on the spheres, for and, are polynomials of degree,, and in momenta.We prove noncommutative integrability of the obtained magnetic flows for any when the systems allow a reduction to the cases with. We conjecture that the restricted magnetic systems on are integrable for all.