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

In this paper, we prove a recent conjecture of [V. Dragović, B. Gajić and B. Jovanović, Integrability of homogeneous exact magnetic flows on spheres, Regul. Chaotic Dyn. 30(4) (2025) 582–597, https://doi.org/10.1134/S1560354725040082 ] stating that the magnetic geodesic flow on the standard sphere [Formula: see text] whose magnetic 2-form is the restriction of a constant 2-form from [Formula: see text] is Liouville integrable. The integrals are quadratic and linear in momenta.