  In [3] (Rend. Lincei Mat. Appl. 26 (2015), 1-10; see also arXiv:1503.08145
[math.DS]) 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.
