Integrability of homogeneous exact magnetic flows on spheres for all dimensions

Establish integrability of the homogeneous exact magnetic flows on the sphere S^{n-1} for all dimensions n and for every skew-symmetric matrix κ ∈ so(n) defining the constant homogeneous magnetic 2-form F = s ∑_{i<j} κ_{ij} dγ_i ∧ dγ_j restricted to S^{n-1}.

Background

The paper studies the motion of a unit mass material point on the sphere S^{n-1} under a constant homogeneous magnetic field given by a 2-form F = s∑{i<j} κ{ij} dγ_i ∧ dγ_j, with κ ∈ so(n). Previous work by the same authors established integrability for specific low-dimensional cases (n=3,4) and later for n=5,6, along with certain noncommutative integrability scenarios.

In their earlier work (DGJ2025), they concluded with a conjecture asserting integrability for all dimensions n and all choices of κ when the system is restricted to S^{n-1}. The present note provides a Lax representation and proves Liouville integrability, thereby resolving that conjecture.