  We have laid out the results of a rigorous theoretical investigation into the
response of electron dressed states, i.e., interacting Floquet states arising
from the off-resonant coupling of Dirac spin-1 electrons in the $\alpha$-$T_3$
model, to external radiation with various polarizations. Specifically, we have
examined the role played by the parameter $\alpha$ that is a measure of the
coupling strength with the additional atom at the center of the honeycomb
graphene lattice and which, when varied, continuously gives a different Berry
phase. We have found that the electronic properties of the $\alpha$ -$T_3$
model (consisting of a flat band and two cones) could be modified depending on
the polarization of the imposed irradiation. We have demonstrated that under
elliptically-polarized light the low-energy band structure of such lattice
directly depends on the valley index $\tau$. We have obtained and analyzed the
corresponding wave functions, their symmetries and the corresponding Berry
phases, and revealed that such phases could be finite even for a dice lattice,
which has not been observed in the absence of the dressing field. This results
lead to possible radiation-generated band structure engineering, as well as
experimental and technological realization of such optoelectronic devices and
photonic crystals.
