No monodromy in the champagne bottle, or singularities of a superintegrable system

Abstract. The three-dimensional champagne bottle system contains no mondromy, despite being entirely composed of invariant two-dimensional champagne bottle systems, each of which posesses nontrivial monodromy. We explain where the monodromy went in the three-dimensional system, or perhaps, where it did come from in the two-dimensional system, by regarding the three-dimensional system not as completely integrable, but as superintegrable (or non-commutatively integrable), and explaining the role of the singularities of its isotropic-coisotropic pair of foliations.