2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/214131We define a large class of integrable nonlinear PDE's, \emph{$k$-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the solution, and show that the problem of isometrically immersing $n$-dimensional Euclidean space into a sphere of dimension $2n-1$ can be addressed via this scheme, producing infinitely many real analytic solutions.21 pages. Minor typographical correctionsDifferential GeometryDynamical Systems53C42; 37J35; 53C35text