Analyzing the relationship between infinite symmetries and $N$-soliton solutions in the AKNS system
This paper investigates the algebraic reduction of the infinite-dimensional symmetries of the Ablowitz-Kaup-Newell-Segur system when restricted to multi-soliton solution. By systematically analyzing, we demonstrate that the entire $K$-symmetry hierarchy collapses into a finite-dimensional module over the field of wave parameters, spanned by elementary center-translation generators. Higher order $K$-symmetries are explicitly reconstructed as linear combinations of these basis vectors. In contrast, $τ$-symmetries resist such decomposition on pure soliton backgrounds, however, they become finite-dimensional once the underlying solution space is extended to the full multi-wave manifold, which carries more independent wave parameters. We construct an explicit basis consisting of four fundamental symmetry vector fields, two lowest $K$-symmetries and two primary $τ$-symmetries, in terms of which all higher $τ$-symmetries are uniquely expressible as linear combinations of these symmetry vector fields. These findings not only clarify the algebraic structure of infinite symmetries on special solution, but also provide an algorithmic framework for deriving exact multi-wave solutions of integrable systems.