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 and prove complete integrability of those systems for any $n$. The integrability is provided via first integrals of degree one and two.

6 pages, 1 figure, a shorten version to appear in Russian Math. Surveys, Vol. 80, No. 5, 2025