On the density of mechanical Lagrangians in T2 without continuous invariant graphs in all supercritical energy levels

We show that the set of C mechanical Lagrangians L(p, v) in T-2 without continuous invariant graphs in all supercritical energy levels is dense in the Cl topology. A mechanical Lagrangian L : TT2 -> R is a function in the tangent space of the torus TT2 given by L(p, v) = 1/2g(v, v) - U(p), where g is a Riemannian metric and U is a smooth potential.