  We construct new solutions of the Faddeev-Skyrme model with a symmetry
breaking potential admitting $S^1$ vacuum. It includes, as a limiting case, the
usual $SO(3)$ symmetry breaking mass term, another limit corresponds to the
potential $m^2 \phi_1^2$, which gives a mass to the corresponding component of
the scalar field. However we find that the spacial distribution of the energy
density of these solutions has more complicated structure, than in the case of
the usual Hopfions, typically it represents two separate linked tubes with
different thicknesses and positions. In order to classify these configurations
we define a counterpart of the usual position curve, which represents a
collection of loops $\mathcal{C}_1, \mathcal{C}_{-1}$ corresponding to the
preimages of the points $\vec \phi = (\pm 1 \mp \mu, 0,0)$, respectively. Then
the Hopf invariant can be defined as $Q= {\rm link}
(\mathcal{C}_1,\mathcal{C}_{-1})$. In this model, in the sectors of degrees
$Q=5,6,7$ we found solutions of new type, for which one or both of these tubes
represent trefoil knots. Further, some of these solutions possess different
types of curves $\mathcal{C}_1$ and $\mathcal{C}_{-1}$.
