  We study the singular locus of solutions to Hamilton-Jacobi equations with a
Hamiltonian independent of $u$. In a previous paper, we proved that the
singular locus is what we call a balanced split locus. In this paper, we find
and classify all balanced split sets, identifying the cases where the only
balanced split locus is the singular locus, and the cases where this doesn't
hold. This clarifies the relationship between viscosity solutions and the more
classical approach of characteristics and shocks.
