  We consider the exact reduced dynamics of a two-level system coupled to a
bosonic reservoir, further obtaining the exact time-convolutionless and
Nakajima-Zwanzig non-Markovian equations of motion. The considered system
includes the damped and undamped Jaynes-Cummings model. The result is obtained
by exploiting an expression of quantum maps in terms of matrices and a simple
relation between the time evolution map and time-convolutionless generator as
well as Nakajima-Zwanzig memory kernel. This non-perturbative treatment shows
that each operator contribution in Lindblad form appearing in the exact
time-convolutionless master equation is multiplied by a different time
dependent function. Similarly, in the Nakajima-Zwanzig master equation each
such contribution is convoluted with a different memory kernel. It appears that
depending on the state of the environment the operator structures of the two
set of equations of motion can exhibit important differences.
