  We consider a spin-boson Hamiltonian which is generalized such that the
Hamiltonians for the system ($\hat{H}_{\cal S}$) and the interaction with the
environment ($\hat{H}_{\rm int}$) do not commute with each other. Considering a
single-mode quantized field in exact resonance with the tunneling matrix
element of the system, we obtain the time-evolution operator for our model.
Using our time-evolution operator we calculate the time-dependent pointer
states of the system and the environment (which are characterized by their
ability not to entangle with each other) for the case that the environment
initially is prepared in the coherent state. We show that our solution for the
pointer states of the system and the environment is valid over a length of time
which is proportional to $\bar{n}$, the average number of bosons in the
environment. We also obtain a closed form for the offdiagonal element of the
reduced density matrix of the system and study the decoherence of the central
system in our model. We show that for the case that the system initially is
prepared in one of its pointer states, the offdiagonal element of the reduced
density matrix of the system will be a \emph{sinusoidal function} with a slow
decaying envelope which is characterized by a decay time proportional to
$\bar{n}$; while it will experience a much faster decoherence, when the system
initially is \emph{not} prepared in one of its initial pointer states.
