  The properties of the $B_c$-meson family ($c\bar b$) are still not well
determined experimentally because the specific mechanisms of formation and
decay remain poorly understood. Unlike heavy quarkonia, i.e. the hidden heavy
quark-antiquark sectors of charmonium ($c\bar c$) and bottomonium ($b\bar b$),
the $B_c$-mesons cannot annihilate into gluons and they are, consequently, more
stable. The excited $B_c$ states, lying below the lowest strong-decay
$BD$-threshold, can only undergo through radiative decays and hadronic
transitions to the $B_c$ ground state, which then decays weakly. As a result of
this, a rich spectrum of narrow excited states below the $BD$-threshold appear,
whose total widths are two orders of magnitude smaller than those of the
excited levels of charmonium and bottomonium. In a different article, we
determined bottom-charmed meson masses using a non-relativistic constituent
quark model which has been applied to a wide range of hadron physical
observables, and thus the model parameters are completely constrained. Herein,
continuing to our study of the $B_c$ sector, we calculate the relevant
radiative decay widths and hadronic transition rates between $c\bar b$ states
which are below $BD$-threshold. This shall provide the most promising signals
for discovering excited $B_c$ states that are below the lowest strong-decay
$BD$-threshold. Finally, our results are compared with other models to measure
the reliability of the predictions and point out differences.
