Connectedness of fibers beyond semitoric systems II: ephemeral singular points

In an earlier paper, we proved the connectedness of the fibers of every $2n$-dimensional integrable system satisfying both: the action extends the action of an $(n-1)$-dimensional torus which has a proper moment map, and every tall singular point is non-degenerate and no such point has a hyperbolic block and connected $T$-stabilizer. Unfortunately, these criteria are fairly restrictive. Our main goal in this paper is to find a larger class of integrable systems that has connected fibers by weakening the non-degeneracy assumption above. To achieve this, we introduce ``ephemeral" degenerate singular points, examples of which have appeared in the literature in the context of both $p \! : \! -q$ resonances and special Lagrangian fibrations. Finally, we construct a family of examples that shows that our main theorem meaningfully extends previous results.