k-SYMMETRIC AKS SYSTEMS AND FLAT IMMERSIONS INTO SPHERES

We define a large class of integrable nonlinear PDE’s, 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.