  In this paper, we consider a time independent $C^2$ Hamiltonian, sa\-tisfying
the usual hypothesis of the classical Calculus of Variations, on a non-compact
connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the
existence of weak KAM solutions, or viscosity solutions, for the associated
Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We
also study the role of the amenability of the group of symmetries to understand
when the several critical values that can be associated with the Hamiltonian
coincide.
