On the dynamics and integrability of the Ziegler pendulum

The paper correctly identifies the k2=0 reduced system’s reversibility and states the existence of two-parameter families of periodic solutions under three parameter regimes, then infers local Jacobi integrability. However, its proofs for Propositions 5–7 use only a symmetry lemma and a crossing argument, without rigorously justifying persistence of transverse crossings and families under small parameter perturbations (no explicit implicit-function or normal form/Lyapunov-center argument). The model supplies a precise reversible-systems proof: it identifies the line of equilibria E_w, computes the linearization with spectrum {0, ± i ω(w)}, and applies the reversible Lyapunov-center mechanism to obtain symmetric periodic orbits and local integrability. One caveat is that the model’s case (k1=0, F<0) implicitly assumes sign(Ā12/Ā22)>0 to claim uniform ellipticity; this mild extra hypothesis should be stated. Net: the model gives a rigorous proof (with a small missing hypothesis) where the paper’s argument is conceptually sound but incomplete.