We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat-Heckman, and extend the main results of our previous note on focus-focus singularities ($\\bbS^1$-action, monodromy, and topological classification) to the degenerate case. We also consider the non-Hamiltonian case, local normal forms, etc.