  In this paper, we consider a class of infinitely degenerate partial
differential systems to obtain the Nash equilibria in the mean field games. The
degeneracy in the diffusion and the Hamiltonian may be different. This feature
brings difficulties to the uniform boundness of the solutions, which is central
to the existence and regularity results. First, from the perspective of the
value function in the stochastic optimal control problems, we prove the
Lipschitz continuity and the semiconcavity for the solutions of the
Hamilton-Jacobi equations (HJE). Then the existence of the weak solutions for
the degenerate systems is obtained via a vanishing viscosity method.
Furthermore, by constructing an auxiliary function, we conclude the regularity
of the viscosity solution for the HJE in the almost everywhere sense.
