  We introduce a new class $\mathbb{G}^n_s$ of generic real analytic potentials
on $\mathbb{T}^n$ and study global analytic properties of natural
nearly-integrable Hamiltonians $\frac12 |y|^2+\varepsilon f(x)$, with potential
$f\in \mathbb{G}^n_s$, on the phase space $\varepsilon = B \times \mathbb{T}^n$
with $B$ a given ball in $\mathbb{R}^n$. The phase space $\mathcal{M}$ can be
covered by three sets: a `non-resonant' set, which is filled up to an
exponentially small set of measure $e^{-c K}$ (where $K$ is the maximal size of
resonances considered) by primary maximal KAM tori; a `simply resonant set' of
measure $\sqrt{\varepsilon} K^a$ and a third set of measure $\varepsilon K^b$
which is `non perturbative', in the sense that the $H$-dynamics on it can be
described by a natural system which is {\sl not} nearly-integrable. We then
focus on the simply resonant set -- the dynamics of which is particularly
interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori)
-- and show that on such a set the secular (averaged) 1 degree-of-freedom
Hamiltonians (labelled by the resonance index $k\in\mathbb{Z}^n$) can be put
into a universal form (which we call `Generic Standard Form'), whose main
analytic properties are controlled by {\sl only one parameter, which is uniform
in the resonance label $k$}.
