The master $T$-operator for vertex models with trigonometric $R$-matrices as a classical $\tau$-function

We apply the recently proposed construction of the master $T$-operator to integrable vertex models and the associated quantum spin chains with trigonometric $R$-matrices. The master $T$-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. It also turns out to be the $\tau$-function of an integrable hierarchy of classical soliton equations in the sense that it satisfies the same bilinear Hirota equations. We characterize the class of solutions of the Hirota equations that correspond to eigenvalues of the master $T$-operator and discuss its relation to the classical Ruijsenaars–Schneider system of particles.