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Degeneracies in the spectrum of an adiabatically transported quantum system are important to determine the geometrical phase factor, and may be interpreted as magnetic monopoles. We investigate the mechanism by which constraints acting on the system, related to local symmetries, can create arbitrarily large monopole charges. These charges are associated with different geometries of the degeneracy. An explicit method to compute the charge as well as several illustrative examples are given.doi:10.1088/0305-4470/36/11/314 fatcat:v7oqwyj625el3id7g6chpig6zy