R-matrices of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

International audienceWe review some of the strategies that can be implemented to infer an$R$-matrix from the knowledge of its Hamiltonian. We apply them to theclassification achieved in arXiv:1306.6303, on three state $U(1)$-invariantHamiltonians solvable by CBA, focusing on models for which the $S$-matrix isnot trivial. For the 19-vertex solutions, we recover the $R$-matrices of the well-knownZamolodchikov--Fateev and Izergin--Korepin models. We point out that thegeneralized Bariev Hamiltonian is related to both main and special branchesstudied by Martins in arXiv:1303.4010, that we prove to generate the sameHamiltonian. The 19-vertex SpR model still resists to the analysis, although weare able to state some no-go theorems on its $R$-matrix. For 17-vertex Hamiltonians, we produce a new $R$-matrix