Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori

We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a \(C^{\infty}\)-smooth Hamiltonian circle action, which is persistent under small integrable \(C^{\infty}\) perturbations. We deduce from this result the structural stability of parabolic orbits and show that they are all smoothly equivalent (in the non-symplectic sense) to a standard model. As a corollary, we obtain similar results for cuspidal tori. Our proof is based on showing that every symplectomorphism of a neighbourhood of a parabolic point preserving the first integrals of motion is a Hamiltonian whose generating function is smooth and constant on the connected components of the common level sets.