  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.
