Starting from a result of Albers-Geiges-Zehmisch giving a contact-geometric interpretation in terms of quaternionic symmetries of the lift of the magnetic geodesic flow from S^2 to S^3. As nice as it would be to generalize to higher projective spaces, the previous proof cannot be copy-pasted as it is (since the unitary cotangent bundle S*CP^n are simply connected for n>1). Does it mean there is no way to get a similar geometric interpretation? I'll (try to) convince you that thanks to the hyperkähler structure on the cotangent bundle of CP^n, hope exists!