Integrability of Homogeneous Exact Magnetic Flows on Spheres

We consider motion of a material point placed in a constant homogeneous magnetic field in \(\mathbb{R}^{n}\) and also motion restricted to the sphere \(S^{n-1}\). While there is an obvious integrability of the magnetic system in \(\mathbb{R}^{n}\), the integrability of the system restricted to the sphere \(S^{n-1}\) is highly nontrivial. We prove complete integrability of the obtained restricted magnetic systems for \(n\leqslant 6\). The first integrals of motion of the magnetic flows on the spheres \(S^{n-1}\), for \(n=5\) and \(n=6\), are polynomials of degree \(1\), \(2\), and \(3\) in momenta. We prove noncommutative integrability of the obtained magnetic flows for any \(n\geqslant 7\) when the systems allow a reduction to the cases with \(n\leqslant 6\). We conjecture that the restricted magnetic systems on \(S^{n-1}\) are integrable for all \(n\).