Quantum Hamiltonians generated by the $\mathrm{R}$-matrix of the five-vertex model

We consider solutions of the RLL-relation with the $\mathrm{R}$-matrix related to the five-vertex model. We show that in the case where the quantum space of the $L$-operator is infinite-dimensional and described the Fock space of quantum oscillator, the solution of the RLL-relation gives the phase model with two external fields. In the case of a two-dimensional quantum space, there exist two solutions each corresponding to the five-vertex model, and their special case, corresponding to the four-vertex model. We also derive explicit expressions for quantum Hamiltonians for inhomogeneous in the external fields systems, both in the finite-dimensional and infinite-dimensional cases.