  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. Our
proof is based on showing that every symplectomorphism of a neighbourhood of a
parabolic point preserving the integrals of motion is Hamiltonian whose
generating function is smooth and constant on the connected components of the
common level sets.
