  We introduce and explore an interacting integrable cellular automaton, the
Fredkin staircase, that lies outside the existing classification of such
automata, and has a structure that seems to lie beyond that of any existing
Bethe-solvable model. The Fredkin staircase has two families of ballistically
propagating quasiparticles, each with infinitely many species. Despite the
presence of ballistic quasiparticles, charge transport is diffusive in the d.c.
limit, albeit with a highly non-gaussian dynamic structure factor. Remarkably,
this model exhibits persistent temporal oscillations of the current, leading to
a delta-function singularity (Drude peak) in the a.c. conductivity at nonzero
frequency. We analytically construct an extensive set of operators that
anticommute with the time-evolution operator; the existence of these operators
both demonstrates the integrability of the model and allows us to lower-bound
the weight of this finite-frequency singularity.
