  In this paper, building on prior joint work by Mallios and Ntumba, we show
that $\mathcal A$-\textit{transvections} and \textit{singular symplectic
}$\mathcal A$-\textit{automorphisms} of symplectic $\mathcal A$-modules of
finite rank have properties similar to the ones enjoyed by their classical
counterparts. The characterization of singular symplectic $\mathcal
A$-automorphisms of symplectic $\mathcal A$-modules of finite rank is grounded
on a newly introduced class of pairings of $\mathcal A$-modules: the
\textit{orthogonally convenient pairings.} We also show that, given a
symplectic $\mathcal A$-module $\mathcal E$ of finite rank, with $\mathcal A$ a
\textit{PID-algebra sheaf}, any injective $\mathcal A$-morphism of a
\textit{Lagrangian sub-$\mathcal A$-module} $\mathcal F$ of $\mathcal E$ into
$\mathcal E$ may be extended to an $\mathcal A$-symplectomorphism of $\mathcal
E$ such that its restriction on $\mathcal F$ equals the identity of $\mathcal
F$. This result also holds in the more general case whereby the underlying free
$\mathcal A$-module $\mathcal E$ is equipped with two symplectic $\mathcal
A$-structures $\omega_0$ and $\omega_1$, but with $\mathcal F$ being Lagrangian
with respect to both $\omega_0$ and $\omega_1$. The latter is the analog of the
classical \textit{Witt's theorem} for symplectic $\mathcal A$-modules of finite
rank.
