  We consider the Cauchy problem for the KdV hierarchy -- a family of
integrable PDEs with a Lax pair representation involving one-dimensional
Schr\"odinger operators -- under a local in time boundedness assumption on the
solution. For reflectionless initial data, we prove that the solution stays
reflectionless. For almost periodic initial data with absolutely continuous
spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet
data evolve according to a Lipschitz Dubrovin-type flow, so the solution is
uniquely recovered by a trace formula. This applies to algebro-geometric
(finite gap) solutions; more notably, we prove that it applies to small
quasiperiodic initial data with analytic sampling functions and Diophantine
frequency. This also gives a uniqueness result for the Cauchy problem on the
line for periodic initial data, even in the absence of Craig-type conditions.
