  We determine the cells, whose existence has been announced by Ocneanu, on all
the candidate nimrep graphs except $\mathcal{E}_4^{(12)}$ proposed by di
Francesco and Zuber for the SU(3) modular invariants classified by Gannon. This
enables the Boltzmann weights to be computed for the corresponding integrable
statistical mechanical models and provide the framework for studying
corresponding braided subfactors to realise all the SU(3) modular invariants as
well as a framework for a new SU(3) planar algebra theory.
