  We study aspects of the hydrodynamics of one-dimensional totally asymmetric
K-exclusion, building on the hydrodynamic limit of Seppalainen (1999). We prove
that the weak solution chosen by the particle system is the unique one with
maximal current past any fixed location. A uniqueness result is needed because
we can prove neither differentiability nor strict concavity of the flux
function, so we cannot use the Lax-Oleinik formula or jump conditions to define
entropy solutions. Next we prove laws of large numbers for a second class
particle in K-exclusion. The macroscopic trajectories of second class particles
are characteristics and shocks of the conservation law for the particle
density. In particular, we extend to K-exclusion Ferrari's result that the
second class particle follows a macroscopic shock in the Riemann solution. The
technical novelty of the proofs is a variational representation for the
position of a second class particle, in the context of the variational coupling
method.
