2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74830A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such ``almost-toric 4-manifolds'' which admits a Hamiltonian $S^1$-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application, we derive a Duistermaat-Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.finally a revision of the 2003 preprint. 29 pages, 8 figuresSymplectic GeometryMathematical Physics53D05;53D20;37J15;37J35;57R45text