KAM Theory for secondary tori

Abstract: In 3 the following result has been announced: Theorem. Consider a real-analytic nearly-integrable mechanical system with potential $f$, namely, a Hamiltonian system with real-analytic Hamiltonian $$H(y,x)=\frac12 \sum_{i=1}^n y_i^2 +\epsilon f(x)\ ,$$ $(y,x)\in{\mathbb R}^n\times{\mathbb T}^n$ being standard action--angle variables. For "general non-degenerate" potentials $f$'s there exists $\epsilon_0,a>0$ such that, if $0<\epsilon<\epsilon_0$, then the Liouville measure of the complementary of $H$-invariant tori is smaller than $\epsilon|\log \epsilon|^a$. In this paper we provide a proof of such result.