Monodromy of Hamiltonian systems with complexity 1 torus actions

We consider the monodromy of n-torus bundles in n degree of freedom integrable Hamiltonian systems with a complexity 1 torus action, that is, a Hamiltonian Tn−1 action. We show that orbits with T1 isotropy are associated to non-trivial monodromy and we give a simple formula for computing the monodromy matrix in this case. In the case of 2 degree of freedom systems such orbits correspond to fixed points of the T1 action. Thus we demonstrate that, given a Tn−1 invariant Hamiltonian H, it is the Tn−1 action, rather than H, that determines monodromy.