  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 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 $q=5$.
