A Lax representation and integrability of homogeneous exact magnetic flows on spheres in all dimensions

Abstract: We consider motion of a material point placed in a constant homogeneous magnetic field restricted to the sphere $S^{n-1}$. We provide a Lax representation of the equations of motion for arbitrary $n$ and prove integrability of those systems in the Liouville sense. The integrability is provided via first integrals of degree one and two.