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Using a recently developed method, based on a generalization of the zerocurvature representation of Zakharov and Shabat, we study the integrabilitystructure in the Abelian Higgs model. It is shown that the model containsintegrable sectors, where integrability is understood as the existence ofinfinitely many conserved currents. In particular, a gauge invariantdescription of the weak and strong integrable sectors is provided. Thepertinent integrability conditions are given by a U(1) generalization of thestandard strong and weak constraints for models with two dimensional targetspace. The Bogomolny sector is discussed, as well, and we find that eachBogomolny configuration supports infinitely many conserved currents. Finally,other models with U(1) gauge symmetry are investigated.Comment: corrected typos, version accepted in J. Phys.

Integrability in theories with localU(1) gauge symmetry