  We consider scalar conservation laws with convex flux and random initial
data. The Hopf-Lax formula induces a deterministic evolution of the law of the
initial data. In a recent article, we derived a kinetic theory and Lax
equations to describe the evolution of the law under the assumption that the
initial data is a spectrally negative Markov process. Here we show that: (i)
the Lax equations are Hamiltonian and describe a principle of least action on
the Markov group that is in analogy with geodesic flow on $SO(N)$; (ii) the Lax
equations are completely integrable and linearized via a loop-group
factorization of operators; (iii) the associated zero-curvature equations can
be solved via inverse scattering. Our results are rigorous for $N$-dimensional
approximations of the Lax equations, and yield formulas for the limit $N \to
\infty$. The main observation is that the Lax equations are a $N \to \infty$
limit of a Markovian variant of the $N$-wave model. This allows us to introduce
a variety of methods from the theory of integrable systems.
