  We study the dynamics of a class of integrable non-Hermitian free-fermionic
models driven periodically using a continuous drive protocol characterized by
an amplitude $g_1$ and frequency $\omega_D$. We derive an analytic, albeit
perturbative, Floquet Hamiltonian for describing such systems using Floquet
perturbation theory with $g_1^{-1}$ being the perturbation parameter. Our
analysis indicates the existence of special drive frequencies at which an
approximately conserved quantity emerges. The presence of such an almost
conserved quantity is reflected in the dynamics of the fidelity, the
correlation functions and the half-chain entanglement entropy of the driven
system. In addition, it also controls the nature of the steady state of the
system. We show that one-dimensional (1D) transverse field Ising model, with an
imaginary component of the transverse field, serves as an experimentally
relevant example of this phenomenon. In this case, the transverse magnetization
is approximately conserved; this conservation leads to complete suppression of
oscillatory features in the transient dynamics of fidelity, magnetization, and
entanglement of the driven chain at special drive frequencies. We discuss the
nature of the steady state of the Ising chain near and away from these special
frequencies, demonstrate the protocol independence of this phenomenon by
showing its existence for discrete drive protocols, and suggest experiments
which can test our theory.
