Connectedness of level sets for non-degenerate integrable systems that extend complexity one torus actions

In this paper we study the connectedness of the level sets of integrable systems that have only non-degenerate singular points and extend complexity one $T$-spaces with proper moment maps. Our main result states that if there are no singular points with a hyperbolic block and connected $T$-stabilizer, then each level set 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.