This paper is concerned with the global topology of solution trajectories to integrable Hamiltonian systems near ``degenerate'' critical points of the energy-momentum map. The present study is restricted to two examples, the Kirchhoff top and the spherical pendulum in an axially symmetric quadratic potential. These are both one-parameter families of integrable Hamiltonian systems with isolated critical values for the energy-momentum map. For one value of the parameter these values coalesce. The topology of the energy surface changes at the critical values and this indicates a nontrivial fibration of phase space by the toral fibres at regular points of the energy-momentum map. Using standard methods of reduction and some neat footwork the authors derive the transformations (monodromy) of period integrals on the fibres as the system is taken around the critical points. It is given by a triangular matrix, \({1\;0 \choose 1\;1}\). At the point of coalescence it is seen to be \({1\;0 \choose 2\;1}\), which is consistent with some numerical work on the quadratic, spherical pendulum.