  In this paper we investigate the mathematical properties of the integrability
of the symmetric six-vertex model towards the view of Algebraic Geometry. We
show that the algebraic variety originated from Baxter's commuting transfer
method is birationally isomorphic to a ubiquitous threefold known as Segre
cubic primal. This relation makes it possible to present the most generic
solution for the Yang-Baxter triple associated to this lattice model. The
respective $\mathrm{R}$-matrix and Lax operators are parametrized by three
independent affine spectral variables.
