Structure of quantum L operators for the R matrix of the XXZ model

For the R matrices of the XXX and XXZ models, a study is made of the L operators, i.e., the monodromy matrices with vacuum and finite-dimensional single-particle subspace. All possible vacuum eigenvalues of the diagonal matrix elements are found for such monodromy matrices. In the case of the R matrix of the XXZ model, the ratio of the vacuum eigenvalues is shown to be $\pi$-periodic with respect to the spectral parameter. All monodromy matrices with one-dimensional single-particle subspace are found.