Equivalent Integrable Metrics on the Sphere with Quartic Invariants - Tsiganov, Andrey V. (Symmetry, Integrability and Geometry: Methods and Applications, 2022)
We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere to a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.