  We investigate the relation between integrability and decoherence in central
spin models with more than one central spin. We show that there is a transition
between integrability ensured by the Bethe ansatz and integrability ensured by
complete sets of commuting operators. This has a significant impact on the
decoherence properties of the system, suggesting that it is not necessarily
integrability or nonintegrability which is related to decoherence, but rather
its type or a change from integrability to nonintegrability.
