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Journal of the London Mathematical Society Advance Access published online on February 2, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm109
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© 2008 London Mathematical Society

k-symmetric AKS systems and flat immersions into spheres

David Brander

Department of Mathematics
Faculty of Science
Kobe University
1-1, Rokkodai
Nada-ku
Kobe 657-8501
Japan

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.

2000 Mathematics Subject Classification 53C42, 37J35 (primary), 53C35 (secondary).

Received February 1, 2006; revised September 14, 2007;
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