  Within the frame of quantum dissipation theory, we develop a new hierarchical
equations of motion theory, combined with the small polaron transformation. We
fully investigate the electron transport of a single attractive impurity system
with the strong electron-phonon coupling in charge- Kondo regime. Numerical
results demonstrate the following facts: (i) The density of states curve shows
that the attraction mechanism results in not only the charge-Kondo resonance
(i.e. elastic pair-transition resonance), but also the inelastic
pair-transition resonances and inelastic cotunneling resonances. These signals
are separated discernibly in asymmetric levels about Fermi level; (ii) The
differential conductance spectrum shows the distinct peaks or wiggles and the
abnormal split of pair-transition peaks under the asymmetric bias; (iii) The
improvement of bath-temperature can enhance the phonon-emission
(absorption)-assisted sequential tunneling and also strengthen the signals of
inelastic pair-transition under the asymmetric bias; (iv) The inelastic
dynamics driven by the ramp-up time-dependent voltage presents clear steps
which can be tailored by the duration-time and bath-temperature; (v) The
linear-response spectrum obtained from the linear-response theory in the
Liouville space reveals the excitation signals of electrons' dynamical
transition. Briefly, the physics caused by attraction mechanism stems from the
double-occupancy or vacant-occupation of impurity system
