  We investigate the Yang-Baxter algebra for $\mathrm{U}(1)$ invariant
three-state vertex models whose Boltzmann weights configurations break
explicitly the parity-time reversal symmetry. We uncover two families of
regular Lax operators with nineteen non-null weights which ultimately sit on
algebraic plane curves with genus five. We argue that these curves admit degree
two morphisms onto elliptic curves and thus they are bielliptic. The associated
$\mathrm{R}$-matrices are non-additive in the spectral parameters and it has
been checked that they satisfy the Yang-Baxter equation. The respective
integrable quantum spin-1 Hamiltonians are exhibited.
