  A class of integrable models, such as the one-dimensional transverse-field
Ising model, respond nonmonotonically to a periodic drive with respect to the
driving parameters and freezes almost absolutely for certain combinations of
the latter. In this paper, we go beyond the two-band structure of the
Ising-like models studied previously and ask whether such unusual nonmonotonic
response and near-absolute freezing occur in integrable systems with a higher
number of bands. To this end, we consider a tight-binding model for bilayer
graphene subjected to an interlayer potential difference. We find that when the
potential is driven periodically, the system responds nonmonotonically to
variations in the driving amplitude $V_0$ and frequency $\omega$ and shows near
absolute freezing for certain values of $V_0/\omega$. However, the freezing
occurs only in the presence of a constant bias in the driving, i.e., when $V=
V'+V_0 \cos{\omega t}$. When $V'=0$, the freezing is switched off for all
values of $V_0/\omega$. We support our numerical results with analytical
calculations based on a rotating wave approximation. We also give a proposal to
realize the driven bilayer system via ultracold atoms in an optical lattice,
where the driving can be implemented by shaking the lattice.
