Singular KAM theory for convex Hamiltonian systems

In this note, we briefly discuss how singular KAM Theory - which was worked out in a previous work by L.B. and L.C. for the mechanical case 12 |y|²+ε f(x) - can be extended to convex real analytic nearly integrable Hamiltonian systems with Hamiltonian in action-angle variables given by h(y)+ε f(x) with h convex and generic f.