Analytic conjugation between planar differential systems and potential systems

The paper’s main statement (Theorem 1.1) matches the candidate solution’s goal and the overall construction strategy via matching period–energy data is sound. However, the paper relies on Lemma 2.1, claiming as an “immediate consequence of the Poincaré Normal Form Theorem” that any analytic center is analytically conjugate to a rotational normal form X(ξ,η)=f(ξ^2+η^2)(η∂ξ−ξ∂η). That justification is not correct as stated: Poincaré–Dulac provides a formal normal form, not an analytic one, in the resonant center case. The lemma itself is true, but it needs a different argument (e.g., constructing analytic action–angle coordinates from an analytic first integral on the period annulus, as done by the model). Aside from this gap, the paper’s series-based inverse-period construction of V and its concluding identification of the two normal forms are coherent. The model’s solution supplies the missing analytic normal-form step and uses a classical Abel inversion to construct V, yielding a complete, rigorous proof. Key elements of the paper used here: Theorem 1.1 statement and set-up, Lemma 2.2 for T(E), the series construction in Lemma 2.3, and the concluding matching of F and G via equality of period functions .