A first integration of some knot soliton models

Recently it has been shown that there exists a sector within the Faddeev-Niemi model for which the equations of motion may be reduced to first order equations. However, no solutions to that sector have been given. It is not even known whether this sector contains topologically nontrivial solutions, at all. Here, we show that two models with analytically known Hopf solitons, namely the Nicole and the Aratyn-Ferreira-Zimerman models, possess sectors which can be integrated to first order partial differential equations. The main result is that these sectors are topologically nontrivial. In fact, all analytically known hopfions belong to them.