  Starting from the fusion rules for the algebra $SO(5)_2$ we construct
one-dimensional lattice models of interacting anyons with commuting transfer
matrices of `interactions round the face' (IRF) type. The conserved topological
charges of the anyon chain are recovered from the transfer matrices in the
limit of large spectral parameter. The properties of the models in the
thermodynamic limit and the low energy excitations are studied using Bethe
ansatz methods. Two of the anyon models are critical at zero temperature. From
the analysis of the finite size spectrum we find that they are effectively
described by rational conformal field theories invariant under extensions of
the Virasoro algebra, namely $\mathcal{W}B_2$ and $\mathcal{W}D_5$,
respectively. The latter contains primaries with half and quarter spin. The
modular partition function and fusion rules are derived and found to be
consistent with the results for the lattice model.
