  The exterior differential system for constant mean curvature (CMC) surfaces
in a 3-dimensional space form is an elliptic Monge-Ampere system defined on the
unit tangent bundle. We determine the infinite sequence of higher-order
symmetries and conservation laws via an enhanced prolongation modelled on a
loop algebra valued formal Killing field. As a consequence we establish
Noether's theorem for the CMC system and there is a canonical isomorphism
between the symmetries and conservation laws.
  A geometric interpretation of the $\mathbb{S}^{1}$-family of associate
surfaces leads to an integrable extension for a non-local symmetry called
spectral symmetry. We show that the corresponding spectral conservation law
exists as a secondary characteristic cohomology class.
  For a compact linear finite type CMC surface of arbitrary genus, we observe
that the monodromies of the associated flat
$\mathfrak{sl}(2,\mathbb{C})$-connection commute with each other. It follows
that a single spectral curve is defined as the completion of the set of
eigenvalues of the entire monodromies. This agrees with the recent result that
a compact high genus linear finite type CMC surface necessarily factors through
a branched covering of a torus.
  We introduce a sequence of Abel-Jacobi maps defined by the periods of
conservation laws. For the class of deformations of CMC surfaces which scale
the Hopf differential by a real parameter, we compute the first order truncated
Picard-Fuchs equation. The resulting formulae for the first few terms exhibit a
similarity with the Griffiths transversality theorem for variation of Hodge
structures.
