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We consider motion of a material point placed in a constant homogeneous magnetic field in R^n and also motion restricted to the sphere S^n-1. While there is an obvious integrability of the magnetic system in 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 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 7 when the systems allow a reduction to the cases with n 6. We conjecture that the restricted magnetic systems on S^n-1 are integrable for all n.
Dragović et al. (Fri,) studied this question.