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We derive the transfer matrix eigenvalues of a three-state vertex model whoseweights are based on a $\mathrm{R}$-matrix not of difference form with spectralparameters lying on a genus five curve. We have shown that the basic buildingblocks for both the transfer matrix eigenvalues and Bethe equations can beexpressed in terms of meromorphic functions on an elliptic curve. We discussthe properties of an underlying spin one chain originated from a particularchoice of the $\mathrm{R}$-matrix second spectral parameter. We presentnumerical and analytical evidences that the respective low-energy excitationscan be gapped or massless depending on the strength of the interactioncoupling. In the massive phase we provide analytical and numerical evidences infavor of an exact expression for the lowest energy gap. We point out that thecritical point separating these two distinct physical regimes coincides withthe one in which the weights geometry degenerate into union of genus onecurves.

The spectrum of a vertex model and related spin one chain sitting in a genus five curve