Irreducible monodromy matrices for the $R$ matrix of the $XXZ$ model and local lattice quantum Hamiltonians

Monodromy matrices with vacuum and finite-dimensional single-particle subspace are considered for the $R$ matrices of the $XXX$ and $XXZ$ models. A natural class of monodromy matrices – irreducible monodromy matrices – is described; for these matrices, the propositions proposed earlier as natural hypotheses are valid. The existence of local Hamiltonians is proved for quantum integrable models on a lattice with irreducible local monodromy matrices.