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In this article we consider integrable systems on manifolds endowed withsingular symplectic structures of order one. These structures are symplecticaway from an hypersurface where the symplectic volume goes either to infinityor to zero in a transversal way (singularity of order one) resulting either ina $b$-symplectic form or a folded symplectic form. The hypersurface where theform degenerates is called critical set. We give a new impulse to theinvestigation of the existence of action-angle coordinates for these structuresinitiated in [KM] and [KMS] by proving an action-angle theorem for foldedsymplectic integrable systems. Contrary to expectations, the action-anglecoordinate theorem for folded symplectic manifolds cannot be presented as acotangent lift as done for symplectic and $b$-symplectic forms in [KM]. Globalconstructions of integrable systems are provided and obstructions for globalexistence of action-angle coordinates are investigated in both scenarios. Thenew topological obstructions found emanate from the topology of the criticalset $Z$ of the singular symplectic manifold. The existence of theseobstructions in turn implies the existence of singularities for the integrablesystem on $Z$.

Integrable systems on singular symplectic manifolds: From local to global