  We present generic conditions for phase band crossings for a class of
periodically driven integrable systems represented by free fermionic models
subjected to arbitrary periodic drive protocols characterized by a frequency
$\omega_D$. These models provide a representation for the Ising and $XY$ models
in $d=1$, the Kitaev model in $d=2$, several kinds of superconductors, and
Dirac fermions in graphene and atop topological insulator surfaces. Our results
demonstrate that the presence of a critical point/region in the system
Hamiltonian (which is traversed at a finite rate during the dynamics) may
change the conditions for phase band crossings that occur at the critical
modes. We also show that for $d>1$, phase band crossings leave their imprint on
the equal-time off-diagonal fermionic correlation functions of these models;
the Fourier transforms of such correlation functions, $F_{\vec k_0}(
\omega_0)$, have maxima and minima at specific frequencies which can be
directly related to $\omega_D$ and the time at which the phase bands cross at
$\vec k = \vec k_0$. We discuss the significance of our results in the contexts
of generic Hamiltonians with $N>2$ phase bands and the underlying symmetry of
the driven Hamiltonian.
