  We analyze the stability under time evolution of complexifier coherent states
(CCS) in one-dimensional mechanical systems. A system of coherent states is
called stable if it evolves into another coherent state. It turns out that a
system can only poses stable CCS if the classical evolution of the variable for
a given complexifier C depends only on z itself and not on its complex
conjugate. This condition is very restrictive in general so that only few
systems exist that obey this condition. However, it is possible to access a
wider class of models that in principle may allow for stable coherent states
associated to certain regions in the phase space by introducing action-angle
coordinates.
