  We extend the inverse spectral transform for the conservative Camassa-Holm
flow on the line to a class of initial data that requires strong decay at one
endpoint but only mild boundedness-type conditions at the other endpoint. The
latter condition appears to be close to optimal in a certain sense for the
well-posedness of the conservative Camassa-Holm flow. As a byproduct of our
approach, we also find a family of new (almost) conservation laws for the
Camassa-Holm equation, which could not be deduced from its bi-Hamiltonian
structure before and which are connected to certain Besov-type norms (however,
in a rather involved way). These results appear to be new even under positivity
assumptions on the corresponding momentum, in which case the conservative
Camassa-Holm flow coincides with the classical Camassa-Holm flow and no
blow-ups occur.
