Integrability of the magnetic geodesic flow on the sphere with a constant 2-form

We prove a recent conjecture of Dragovic et al arXiv2504.20515 stating that the magnetic geodesic flow on the standard sphere $S^n\subset {\mathbb{R}}^{n+1}$ whose magnetic 2-form is the restriction of a constant 2-form from ${\mathbb{R}}^{n+1}$ is Liouville integrable. The integrals are quadratic and linear in momenta.