Bifurcations of Magnetic Geodesic Flows on Surfaces of Revolution

The candidate solution reconstructs (f, Λ) from a C∞ curve γ = (a, −1, k) satisfying (i*)–(iv*) by the same projective-duality mechanism proved in the paper: Pγ and PΓ are dual, the tangent line at Pγ(r) is f(r) a − Λ(r) + k = 0, and the reconstruction formulas f = −k′/a′ and Λ = k + af hold off a finite set, with uniqueness, equivariance, and the R-map (h, k) ↦ (±√(2h), −1, k) ensuring sign via k′a′ ≤ 0. These are exactly Lemmas 5.12–5.13 and Theorem 5.18, together with Lemma 5.16 and Corollary 5.17 in the paper. Apart from a minor sign slip in the identity κγ = −f′(a′)² (the model writes +), the reasoning matches the paper’s logic and hypotheses, including Assumption 2.4 and the cusp/inflection structure of good projective curves. Hence both are correct and essentially the same proof strategy. See Lemma 5.12, Lemma 5.13, Lemma 5.16, Assumption 2.4, and Theorem 5.18 in the paper for the precise statements and proofs .