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