We study the Yang-Baxter equation for the $R$-matrices of the six-vertex
model. We analyze the solutions and give new parametrizations of the
Yang-Baxter equation. In particular, we find the maximal commutative families
of parametrized solutions which generalize the $R$-matrices from the affine
quantum (super)-groups. Then we give a new parametrization of the Yang-Baxter
equation by a groupoid of non-free-fermionic matrices. In the appendix, we
study the general algebraic structure of the solutions of the Yang-Baxter and
formulate a conjecture that extends the conjecture by Brubaker, Bump, and
Friedberg that the composition law on the Yang-Baxter solutions is always
associative.
