  We consider the Ising chain driven by oscillatory transverse magnetic fields.
For certain parameter regimes, we reveal a hidden integrable structure in the
problem, which allows access to the \textit{exact time-evolution} in this
driven quantum system. We compute time-evolved one- and two-point functions
following a quench that activates the driving. It is shown that this model does
not heat up to infinite temperature, despite the absence of energy
conservation, and we further discuss the generalization to a family of driven
Hamiltonians that do not suffer heating to infinite temperature, despite the
absence of integrability and disorder. The particular model studied in detail
also presents a route for realising exotic physics (such as the E8 perturbed
conformal field theory) via driving in quantum chains that could otherwise
never realise such behaviour. In particular we numerically confirm that the
ratio of the meson excitations masses is given by the golden ratio.
