Connectedness of fibers beyond semitoric systems I: the non-degenerate case

In this paper we study the connectedness of the fibers of integrable systems that extend complexity one $T$-spaces with proper moment maps, assuming that every tall singular point is non-degenerate. Our main result states that if there are no tall singular points with a hyperbolic block and connected $T$-stabilizer, then each fiber is connected. Moreover, we prove that the above condition is necessary if either some reduced space is simply connected or the moment map for the integrable system is generic in a natural sense.