  We obtain the exact scale invariant scattering solutions for two-dimensional
field theories with replicated permutational symmetry $\mathbb{S}_q$. After
sending to zero the number of replicas they correspond to the renormalization
group fixed points of the $q$-state Potts model with quenched disorder. We find
that all solutions with non-zero disorder possess $q$-independent sectors,
pointing to superuniversality (i.e. symmetry independence) of some critical
exponents. The solution corresponding to the random bond ferromagnet, for which
disorder vanishes as $q\to 2$, allows for superuniversality of the correlation
length exponent $\nu$ [PRL 118 (2017) 250601]. Of the two solutions which are
strongly disordered for all values of $q$, one is completely $q$-independent
and accounts for the zero-temperature percolation fixed point of the randomly
bond diluted ferromagnet. The other is the main candidate to describe the
Nishimori-like fixed point of the Potts model with $\pm J$ disorder, and leaves
room for superuniversality of the magnetic exponent $\eta$, a possibility not
yet excluded by available numerical data.
