  The geometric theory of pseudo-differential and Fourier Integral Operators
relies on the symplectic structure of cotangent bundles. If one is to study
calculi with some specific feature adapted to a geometric situation, the
corresponding notion of cotangent bundle needs to be adapted as well and leads
to spaces with a singular symplectic structure. Analysing these singularities
is a necessary step in order to construct the calculus itself.
  In this article we provide some new insights into the symplectic structures
arising from asymptotically Euclidean manifolds. In particular, we study the
action of the Poisson bracket on $SG$-pseudo-differential operators and define
a new class of singular symplectomorphisms, taking into account the geometric
picture. We then consider this notion in the context of the characterisation of
order-preserving isomorphisms of the $SG$-algebra, and show that these are in
fact given by conjugation with a Fourier Integral Operator of $SG$-type.
