  We continue our study of tempered oscillatory integrals $I_\varphi(a)$, here
investigating the link with a suitable symplectic structure at infinity, which
we describe in detail. We prove adapted versions of the classical theorems,
which show that tempered distributions of the type $I_\varphi(a)$ are indeed
linked to suitable Lagrangians extending to infinity, that is, extending up to
the boundary and in particular the corners of a compactification of
$T^*\mathbb{R}^d$ to $\mathbb{B}^d\times\mathbb{B}^d$. In particular, we show
that such Lagrangians can always be parametrized by non-homogeneous, regular
phase functions, globally defined on some $\mathbb{R}^d\times\mathbb{R}^s$. We
also state how two such phase functions parametrizing the same Lagrangian may
be considered equivalent up to infinity.
