  The zigzag model is a relativistic $N$-body system arising in the high energy
limit of the worldsheet scattering in adjoint two-dimensional QCD. We prove
classical Liouville integrability of this model by providing an explicit
construction of $N$ charges in involution. Furthermore, we also prove that the
system is maximally superintegrable by constructing $N-1$ additional
independent charges. All of these charges are piecewise linear functions of
coordinates and momenta. The classical time delays are determined algebraically
from this integrable structure. The resulting $S$-matrix is the same as in the
$N$-particle subsector of a massless $T\bar{T}$ deformed fermion.
