An averaging method is developed in order to determine analytically the magnetic structure of a system with symmetry broken by small perturbations. In particular. Poincare maps are obtained for toroidal helical fields using the typical parameters of the Brazilian Tokamak TBR-1. The technique is fairly general as the small parameter used in the construction of this theory is the original symmetry-breaking perturbation parameter; it is applicable in analyses of fields in more compact devices or with cross-sections with any shape as well. Unless the unperturbed system is doubly symmetric (circular cylinder) a single helical perturbation mode (m, n) can excite many other modes. The coupling between any two modes arises in a very natural way.