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We investigate the Yang-Baxter algebra for $\mathrm{U}(1)$ invariantthree-state vertex models whose Boltzmann weights configurations breakexplicitly the parity-time reversal symmetry. We uncover two families ofregular Lax operators with nineteen non-null weights which ultimately sit onalgebraic plane curves with genus five. We argue that these curves admit degreetwo morphisms onto elliptic curves and thus they are bielliptic. The associated$\mathrm{R}$-matrices are non-additive in the spectral parameters and it hasbeen checked that they satisfy the Yang-Baxter equation. The respectiveintegrable quantum spin-1 Hamiltonians are exhibited.

Integrable three-state vertex models with weights lying on genus five curves