Classifying Potts critical lines

We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points invariant under the permutational symmetry S_{q} in two dimensions, and we show how one of these scattering solutions describes the ferromagnetic and square lattice antiferromagnetic critical lines of the q-state Potts model. Other solutions we determine should correspond to new critical lines. In particular, we obtain that a S_{q}-invariant fixed point can be found up to the maximal value q=(7+sqrt[17])/2. This is larger than the usually assumed maximal value 4 and leaves room for a second-order antiferromagnetic transition at q=5.