This is a part of a further undertaking to affirm that most of
classical module theory may be retrieved in the framework of Abstract Differential
Geometry (`a la Mallios). More precisely, within this article, we study some
defining basic concepts of symplectic geometry on free A-modules by focussing
in particular on the group sheaf of A-symplectomorphisms, where A is assumed
to be a torsion-free PID C-algebra sheaf. The main result arising hereby is that
A-symplectomorphisms locally are products of symplectic transvections, which is
a particularly well-behaved counterpart of the classical result.