  We derive the transfer matrix eigenvalues of a three-state vertex model whose
weights are based on a $\mathrm{R}$-matrix not of difference form with spectral
parameters lying on a genus five curve. We have shown that the basic building
blocks for both the transfer matrix eigenvalues and Bethe equations can be
expressed in terms of meromorphic functions on an elliptic curve. We discuss
the properties of an underlying spin one chain originated from a particular
choice of the $\mathrm{R}$-matrix second spectral parameter. We present
numerical and analytical evidences that the respective low-energy excitations
can be gapped or massless depending on the strength of the interaction
coupling. In the massive phase we provide analytical and numerical evidences in
favor of an exact expression for the lowest energy gap. We point out that the
critical point separating these two distinct physical regimes coincides with
the one in which the weights geometry degenerate into union of genus one
curves.
