  Floquet engineering of topological phase transitions driven by a
high-frequency time-periodic field is a promising approach to realizing new
topological phases of matter distinct from static states. Here, we
theoretically investigate Floquet engineering topological phase transitions in
the quantum spin Hall $\alpha$-$T_{3}$ system driven by an off-resonant
circularly polarized light. In addition to the quantum spin (anomalous) Hall
insulator phase with multiple helical (chiral) edge states, spin-polarized
topological metallic phases are expected, where the bulk topological band gap
of one spin sub-band overlaps with the other gapless spin sub-band. Moreover,
with a staggered potential, the topological invariants of the system depend on
whether the middle band is occupied because of the breaking of particle-hole
symmetry. Our work highlights the significance of Floquet engineering in
realizing new topological phases in the $\alpha$-$T_{3}$ lattice.
