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We remind that stands for an empty site.
Be careful that the solutions in Ref. 13 have to be braided (i.e., multiplied by the permutation P on the left) to get .

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Using the nested coordinate Bethe ansatz, we study 3-state Hamiltonians with 33 non-vanishing entries, or 33-vertex models, where only one global charge with degenerate eigenvalues exists and each site possesses three internal degrees of freedom. In the context of Markovian processes, they correspond to diffusing particles with two possible internal states which may be exchanged during the diffusion (transmutation). The first step of the nested coordinate Bethe ansatz is performed providing the eigenvalues in terms of rapidities. We give the constraints ensuring the consistency of the computations. These rapidities also satisfy Bethe equations involving 4 × 4 -matrices, solutions of the Yang–Baxter equation which implies new constraints on the models. We solve them allowing us to list all the solvable 33-vertex models.


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