NONINTEGRABILITY OF TIME-PERIODIC PERTURBATIONS OF SINGLE-DEGREE-OF-FREEDOM HAMILTONIAN SYSTEMS NEAR THE UNPERTURBED HOMO- AND HETEROCLINIC ORBITS

The paper’s Theorem 1.2 proves that if the Melnikov function M(θ) is not constant under (A1)–(A3), then the autonomous system (1.6) is not real‑meromorphically integrable near Γ̂, by showing that the monodromy subgroup of the variational equation (reduced to a 3×3 block) is non‑commutative; this sits inside the differential Galois group, so Ayoul–Zung implies nonintegrability . The candidate solution argues via a triangular NVE, computes a “scattering” connection giving a unipotent element, and combines it with a diagonal subgroup from the oscillator block to exhibit a non‑abelian identity component, again invoking Ayoul–Zung. Both routes hinge on the same Melnikov coefficients (equivalently, M̂ℓ ≠ 0) . The paper’s approach is via monodromy on a Riemann surface and explicit conjugation formulas; the candidate’s is a direct connection/torus argument. Aside from minor rigor gaps in justifying that the “scattering” connection lies in the Galois group, the model’s reasoning matches the paper’s conclusion.