2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/187216Commuting transfer matrices of $U_{q}(X_{r}^{(1)})$ vertex models obey the functional relations which can be viewed as an $X_{r}$ type Toda field equation on discrete space time. Based on analytic Bethe ansatz we present, for $X_{r}=D_{r}$, a new expression of its solution in terms of determinants and Pfaffians.Latex, 14 pages, ioplppt.sty and iopl12.sty assumedHigh Energy Physics - TheoryCondensed MatterExactly Solvable and Integrable Systemstext