  We discuss the $F$-matrices associated to the $R$-matrix of a general
$N$-state vertex model whose statistical configurations encode $N-1$ U(1)
symmetries. The factorization condition is shown for arbitrary weights being
based only on the unitarity property and the Yang-Baxter relation satisfied by
the $R$-matrix. Focusing on the N=3 case we are able to conjecture the
structure of some relevant twisted monodromy matrix elements for general
weights. We apply this result providing the algebraic expressions of the domain
wall partition functions built up in terms of the creation and annihilation
monodromy fields. For N=3 we also exhibit a $R$-matrix whose weights lie on a
del Pezzo surface and have a rather general structure.
