Monodromy in non-integrable systems on certain compact classical phase spaces

On the example of bending vibrational polyads of the acetylene molecule (C₂H₂) in the approximation of the $1:1:1:1$ resonant oscillator with axial symmetry, whose geometry is similar to the n-shell approximation of the perturbed hydrogen atom, we show how remaining invariant tori of the underlying classical non-integrable system form a nontrivial continuous family with monodromy. We read this monodromy off the quantum energy spectrum which was observed experimentally by spectroscopists, and we uncover its origins through the particular topology, geometry, and symmetry. We explain how monodromy characterizes the chaotic region surrounded by the tori. We detail the explicit correspondence between the bending polyads of C₂H₂ and the n-shells of the hydrogen atom, and uncover the dynamical SO(3) symmetry of the bending polyads and the corresponding spherically localized vibrational states.