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In the present paper, we consider a smooth $C^\infty$ symplecticclassification of Lagrangian fibrations near cusp singularities, parabolicorbits and cuspidal tori. We show that for these singularities as well as foran arrangement of singularities known as a flap, which arises in the integrablesubcritical Hamiltonian Hopf bifurcation, the action variables form a completeset of $C^\infty$ symplectic invariants. We also give a symplecticclassification for parabolic orbits in the real-analytic case. Namely, we provethat a complete symplectic invariant in this case is given by a real-analyticfunction germ in two variables. Additionally, we construct several symplecticnormal forms in the $C^\infty$ and/or real-analytic categories, includingreal-analytic right and right-left symplectic normal forms for parabolicorbits.

$C^\infty$ symplectic invariants of parabolic orbits and flaps in integrable Hamiltonian systems