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

*2021, Volume 26, Number 6, pp. 732-741*

Author(s):

**Kudryavtseva E. A., Martynchuk N. N.**

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.

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