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

The paper proves Liouville integrability for magnetic geodesic flow on S^n with a constant ambient 2-form by (i) a global gauge/fiber-translation that converts the twisted symplectic form to the canonical one, yielding a canonical Hamiltonian H_pert = 1/2 g^{ij}(p_i - σ_i)(p_j - σ_j) (eq. (3)), and (ii) identifying H_pert as the Hamiltonian of a degenerate Neumann system plus a commuting linear integral generated by blockwise rotations, then assembling ⌊n/2⌋ quadratic integrals with m = ⌊(n+1)/2⌋ linear integrals to obtain n commuting, a.e.-independent integrals, with H lying in their span. This is exactly the model’s outline: the same gauge map, the same identification with a block-degenerate Neumann system, the same use of Uhlenbeck-type quadratics plus commuting rotational momenta, and the same count of quadratic vs. linear integrals. The paper grounds the degeneracy step via a careful passage-to-the-limit/Benenti condition argument and an isomorphism to curvature-type tensors, ensuring commutativity and independence in the degenerate case; the model sketches this step heuristically via standard limits of Uhlenbeck integrals. Minor normalization/sign discrepancies in the model (e.g., missing 1/2 in the linear term and 1/8 vs. 1/2 in the potential) do not affect the integrability conclusion. Key steps and claims in the paper: setup of Ω_pert and H (eqs. (1)–(2)), the global σ and the translation p ↦ p + σ yielding eq. (3) and the equivalence of flows (Fact 2.1), canonical normal form of the ambient 2-form and the concrete σ (eqs. (5)–(7)), identification of the linear and potential terms (eq. (9)) and hence the Neumann + linear decomposition (and the displayed relation H_pert = H ± (1/2)∑ α_i M_{2i-1,2i}), followed by the explicit degenerate Uhlenbeck family (eqs. (15)–(17)) and the commuting-with-rotations/limit argument that yields the needed n integrals (Cor. 2.5 and §2.4). These align with the model’s construction and claims. See the paper’s equations and statements cited above for details: equations (1)–(3) and Fact 2.1 for the gauge equivalence, (5)–(7), (9) for the linear/potential identification, and the Neumann/Uhlenbeck constructions and limit argument in §2.3–§2.4 (including Cor. 2.5).