  We show that every global viscosity solution of the Hamilton-Jacobi equation
associated with a convex and superlinear Hamiltonian on the cotangent bundle of
a closed manifold is necessarily invariant under the identity component of the
group of symmetries of the Hamiltonian (We prove that this group is a compact
Lie group). In particular, every Lagrangian section invariant under the
Hamiltonian flow is also invariant under this group.
