  We have found a family of solvable nineteen vertex model with statistical
configurations invariant by the time reversal symmetry within a systematic
study of the respective Yang-Baxter relation. The Boltzmann weights sit on a
degree seven algebraic threefold which is shown birationally equivalent to the
three-dimensional projective space. This permits to write parameterized
expressions for both the transition operator and the $\mathrm{R}$-matrix
depending on three independent affine spectral parameters. The Hamiltonian
limit tells us that the azimuthal magnetic field term is connected with the
asymmetry among two types of spectral variables. The absence of magnetic field
defines a physical submanifold whose geometrical properties are remarkably
shown to be governed by a quartic $\mathrm{K}3$ surface. This expands
considerably the class of irrational manifolds that could emerge in the theory
of quantum integrable models.
