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 $$n$$ .