Three-state chiral Potts models in two dimensions: integrability and symmetry

Symmetry and integrability relations for the three-state chiral Potts model on two-dimensional lattices are reviewed. Our detailed and systematic analysis leads to new integrability varieties for specific chiral Potts models including new integrable points for the standard antiferromagnetic Potts model. We also describe a three-state vertex-model representation of the chiral Potts model, and write down the associated fundamental invariants for three-coordinated lattices.