  We propose an extension of the structure equation for constant mean curvature
(CMC) surfaces in a three dimensional Riemannian space form to the associated
CMC hierarchy of evolution equations by the higher-order commuting symmetries.
Via the canonical formal Killing field, considered as an infinitely prolonged
and loop algebra valued Gau\ss$\,$ map, the CMC hierarchy is obtained by the
assembly of a pair of Adler-Kostant-Symes bi-Hamiltonian hierarchies to the
original CMC system. The infinite sequence of higher-order conservation laws of
the CMC system admits the corresponding extension, and we find a formula for
the generating series of the representative 1-forms. We also introduce a class
of generalized (complexified) CMC surfaces as the phase space of the CMC
hierarchy.
