  We propose an extension of the differential system for constant mean
curvature (CMC) surfaces in a three dimensional space form to an associated
hierarchy of evolution equations by the higher-order commuting symmetries. The
infinite sequence of higher-order conservation laws of CMC surfaces admit the
corresponding extension to the conservation laws of the entire hierarchy. A
class of generalized CMC surfaces are introduced as the phase space of the
hierarchy. Via a canonical dressing transformation, we define a wave function
for CMC hierarchy. This leads to a pair of additional non-local formal Killing
fields and the associated spectral (Virasoro) Killing field.

Differential Geometry