We initiate a novel application of the quantum inverse scattering method for the 20-vertex model, building upon seminal work from Faddeev and Takhtajan on the study of Hamiltonian systems. In comparison to a previous work of the author in late 2023 which characterized integrability of a Hamiltonian flow for the 6-vertex model from integrability of inhomogeneous limit shapes, formalized in a work of Keating, Reshetikhin and Sridhar, notions similar to those of integrability can be realized for the 20-vertex model by studying new classes of higher-dimensional L-operators. Such L-operators provided by Boos and colleagues have algebraic, combinatorial, and geometric, qualities, all of which impact leading order approximations of correlations, products of L-operators, the transfer matrix, and the quantum monodromy matrix.