  Chains of interacting non-Abelian anyons with local interactions invariant
under the action of the Drinfeld double of the dihedral group $D_3$ are
constructed. Formulated as a spin chain the Hamiltonians are generated from
commuting transfer matrices of an integrable vertex model for periodic and
braided as well as open boundaries. A different anyonic model with the same
local Hamiltonian is obtained within the fusion path formulation. This model is
shown to be related to an integrable fusion interaction round the face model.
Bulk and surface properties of the anyon chain are computed from the Bethe
equations for the spin chain. The low energy effective theories and operator
content of the models (in both the spin chain and fusion path formulation) are
identified from analytical and numerical studies of the finite size spectra.
For all boundary conditions considered the continuum theory is found to be a
product of two conformal field theories. Depending on the coupling constants
the factors can be a $Z_4$ parafermion or a $\mathcal{M}_{(5,6)}$ minimal
model.
