2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/93199We examine applications of polynomial Lie algebras $sl_{pd}(2)$ to solve physical tasks in $G_{inv}$-invariant models of coupled subsystems in quantum physics. A general operator formalism is given to solve spectral problems using expansions of generalized coherent states, eigenfunctions and other physically important quantities by power series in the $sl_{pd}(2)$ coset generators $V_{\pm}$. We also discuss some mappings and approximations related to the familiar $sl(2)$ algebra formalism. On this way a new closed analytical expression is found for energy spectra which coincides with exact solutions in certain cases and, in general, manifests an availability of incommensurable eigenfrequencies related to a nearly chaotic dynamics of systems under study.8 pages, LATEXQuantum Physicstext