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 provecomplete 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 \(n\) .
