Integrability and Hopf solitons in models with explicitly broken O(3) symmetry

A wide class of models, built of the three component unit vector field living in the (3 + 1) Minkowski space-time, which explicitly break global O(3) symmetry are discussed. The symmetry breaking occurs due to the so-called dielectric function multiplying a standard symmetric term. Integrability conditions are found. Moreover, for some particular forms of the Lagrangian exact toroidal solutions with any Hopf index are obtained. It is proved that such a symmetry breaking influences the shape of the solitons whereas the energy as well as the Hopf index remain unchanged. (orig.)