Balanced split sets and Hamilton-Jacobi equations

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.