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 $n$.