In At first an extension of famous After that vanishing viscosity limit for Hamilton-Jacobi PDE with non-convex is investigated and a new method to augment the standard viscosity solution approach is presented. The main idea is to introduce a solution of the adjoint of the formal linearization, and then to integrate by parts with respect to the density . This introduction is directly iired by weak KAM theory. These tools enable us to understand the precise nature of the vanishing viscosity and obtain a new representation formula for viscosity solutions of nonconvex Hamilton–Jacobi PDE. This representation formula is actually a generation of Hopf's formula to non-convex functions. We also explain some new and basic results about characteristic curves. The align=center F(x,u,p)=0, x D (1) Subjected to some boundary (initial) conditions specified on a hyper surface M : u(x)=w(x) , x M (2) In many applications, and first of all in optimal control and differential games, non-smooth (or discontinuous) functions have to be treated as the solutions to the PDE ( 1). So there are some kinds of singular surfaces that solutions of equations face with some singularities. In the last part of thesis generalized characteristics for Hamilton–Jacobi type PDE are presented. In particular solutions of these ODES are used to deduce Information about the properties of especial kind of these singular surfaces what is called equivocal surfaces. Some new and simple derivations, with attention paid to the geometric and analytic properties of equivocal curve (n=1) and surfaces (n