Polynomial Lie algebras sl$_{pd}$(2) in action: smooth sl(2) mappings and approximations

We 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