Equivalent integrable metrics on the sphere with quartic invariants

The paper’s Proposition 1 states the map ρ: yi = sqrt(bi/H) pxi together with the orthogonality constraint xipxi + yipyi = 0 preserves the Dirac–Poisson bracket on T*Sn−1, the constraints Σx2=1 and Σxipi=0, and the quadratic Hamiltonian H=Σ bi p^2; it also derives ρ(T)=Σ bi ci pxi^2 + H Σ ai bi^{-1} x_i^2 when applied to a second quadratic geodesic Hamiltonian T (all asserted as a straightforward verification). The candidate solution’s Part A–B explicitly performs these bracket computations and substitutions (matching the paper’s claims), and confirms the preservation of constraints and of H, as in the text establishing (2.3)–(2.5) . For n=3 and bi=1, the paper shows that T has a quadratic integral J=√(w(x))H while ρ(T) has no quadratic polynomial integral but does have the quartic integral ρ(J2); this is asserted and exhibited around (2.6)–(2.7) and the discussion following them, in agreement with Part C of the solution (the solution also gives a natural proof idea via pushforward/pullback under the Poisson automorphism ρ) . The paper’s Proposition 2 introduces σ=ρ(c)∘ρ(b) with the identities xipxi = x̃ip̃xi and (bi/H) pxi^2 = (ci/Ĥ) p̃xi^2 (its printed layout in (2.8) is compressed, but the intended relation is clear), and states that σ is canonical . The candidate’s Part D matches this construction and uses it to argue the equivalence of three geodesic metrics H(1), H(2), H(3) written in elliptic coordinates (the paper presents these three cases and proves their equivalence under σ in Proposition 7) . Minor issues in the candidate solution include a few sign slips in intermediate Poisson-derivative identities (which cancel in the final bracket computations) and an extra square on a factor in the squared identity relating momenta after σ; the paper’s own presentation also omits detailed calculations and compresses some formulas. Overall, the constructions and conclusions coincide and the proof strategies are essentially the same.