  We provide a class of quantum evolution beyond Markovian semigroup. This
class is governed by a hybrid Davies like generator such that dissipation is
controlled by a suitable memory kernel and decoherence by standard GKLS
generator. These two processes commute and both of them commute with the
unitary evolution controlled by the systems Hamiltonian. The corresponding
memory kernel gives rise to semi-Markov evolution of the diagonal elements of
the density matrix. However, the corresponding evolution needs not be
completely positive. The role of decoherence generator is to restore complete
positivity. Hence, to pose the dynamical problem one needs two processes
generated by classical semi-Markov memory kernel and purely quantum decoherence
generator. This scheme is illustrated for a qubit evolution.
