Singular KAM Theory for Convex Hamiltonian Systems

In this note, we briefly discuss how the singular KAM theory of [7] — which was worked out for the mechanical case $\frac12 |y|^2+\varepsilon f(x)$ — can be extended to convex real-analytic nearly integrable Hamiltonian systems with Hamiltonian in action-angle variables given by $h(y)+\varepsilon f(x)$ with $h$ convex and $f$ generic.