K-symmetric AKS systems and flat immersions in spheres

We define a large class of integrable nonlinear PDEs, k-symmetric AKS systems, with solutions that 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.