k‐symmetric AKS systems and flat immersions into 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.