This paper explores the regularity properties of an inverse spectral transform for
Hilbert–Schmidt Hankel operators on the unit disc. This spectral transform plays the
role of action-angle variables for an integrable infinite dimensional Hamiltonian system:
the cubic Szegő equation. We investigate the regularity of functions on the tori
supporting the dynamics of this system, in connection with some wave turbulence
phenomenon, discovered in a previous work and due to relative small gaps between
the actions. We revisit this phenomenon by proving that generic smooth functions
and a
dense set of irregular functions do coexist on the same torus. On the other hand, we
establish some uniform analytic regularity for tori corresponding to rapidly
decreasing actions which satisfy some specific property ruling out the phenomenon of
small gaps.