  We examine two exactly solvable models of decoherence -- a central
spin-system, (i) with and (ii) without a self--Hamiltonian, interacting with a
collection of environment spins. In the absence of a self--Hamiltonian we show
that in this model (introduced some time ago to illustrate environment--induced
superselection) generic assumptions about the coupling strengths can lead to a
universal (Gaussian) suppression of coherence between pointer states. On the
other hand, we show that when the dynamics of the central spin is dominant a
different regime emerges, which is characterized by a non--Gaussian decay and a
dramatically different set of pointer states. We explore the regimes of
validity of the Gaussian--decay and discuss its relation to the spectral
features of the environment and to the Loschmidt echo (or fidelity).
