frequency $\omega_*$. Assuming that $H_\varepsilon$ is sufficiently smooth and $\varepsilon$ sufficiently small, under the Kolmogorov non degeneracy condition (a generic condition on $h$, see Definition \[defK\] below) the classical KAM theorem asserts that most of the invariant tori of the integrable system persist under perturbation. More precisely, let $\Omega$ be the image by $\nabla h$ of $\bar{D}$ and, fixing $0<\gamma \leq 1$, $\tau > n-1$, let $$\label{def.Omega}
\Omega_{\gamma,\tau}:= \left\{  \omega \in \Omega \, | \, d(\omega,\partial \Omega) \geq \gamma, \; \omega \in {\rm Dio}_{\gamma,\tau} \right\}.$$ Observe that since $\tau>n-1$, the complement of $\Omega_{\gamma,\tau}$ in $\Omega$ has a measure of order $\gamma$. The KAM theorem asserts that if one assumes that $\varepsilon \leq c \gamma^2$ where $c$ is some positive constant that depends on $h$ and $f$, then there exists a set $$\mathcal{K}=\bigcup_{\omega_* \in \Omega_{\gamma,\tau}}\mathcal{T}_{\omega_*}^\varepsilon \subseteq \T^n \times D$$ whose complement has a Lebesgue measure of order at most $\gamma$, and which consists of Lagrangian Diophantine tori $\mathcal{T}_{\omega_*}^\varepsilon$ invariant by $H_\varepsilon$ that converge to $\mathcal{T}_{\omega_*}$ as $\varepsilon$ goes to zero. For real-analytic systems, this is a classical result (see [@Pos01] and references therein) and for Gevrey smooth systems, this result is due to Popov ([@Pop04]). Our second theorem states that for 