Novikov-Veselov Symmetries of the Two-Dimensional O(N) Sigma Model

We show that Novikov-Veselov hierarchy provides a complete family of commuting symmetries two-dimensional O(N) sigma model. In the first part paper we use these to prove Fermi spectral curve for double-periodic model is algebraic. Thus, our previous construction complexified harmonic maps in case irreducible curves complete. second generalize reducible and it gives conformal even-dimensional spheres. Remarkably, solutions are parameterized by turning points elliptic Calogero-Moser system.