  We study the dynamics of a bulk deterministic Floquet model, the Rule 201
synchronous one-dimensional reversible cellular automaton (RCA201). The system
corresponds to a deterministic, reversible, and discrete version of the PXP
model, whereby a site flips only if both its nearest neighbours are unexcited.
We show that the RCA201/Floquet-PXP model exhibits ballistic propagation of
interacting quasiparticles - or solitons - corresponding to the domain walls
between non-trivial three-fold vacuum states. Starting from the quasiparticle
picture, we find the exact matrix product state form of the non-equilibrium
stationary state for a range of boundary conditions, including both periodic
and stochastic. We discuss further implications of the integrability of the
model.
