  Continuing the previous work, we propose a further extension of the structure
equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro
algebra of non-local symmetries. Via a canonical dressing transformation, we
first define a wave function for the CMC hierarchy. This leads to a pair of
additional formal Killing fields, and the corresponding spectral Killing field
is defined by a purely algebraic formula up to an integrable extension. The
extended CMC hierarchy is obtained by packaging these data into the associated
affine Kac-Moody algebra valued Killing fields. The log of tau function of the
extended CMC hierarchy is defined as the central component of the affine
extension of the spectral Killing field. We give a closed formula for the tau
function in terms of the determinant of the spectral Killing field.
