  Let $[A]: Y'=AY$ with $A\in \mathrm{M}_n (k)$ be a differential linear
system. We say that a matrix $R\in {\cal M}_{n}(\bar{k})$ is a {\em reduced
form} of $[A]$ if $R\in \mathfrak{g}(\bar{k})$ and there exists $P\in GL_n
(\bar{k})$ such that $R=P^{-1}(AP-P')\in \mathfrak{g}(\bar{k})$. Such a form is
often the sparsest possible attainable through gauge transformations without
introducing new transcendants. In this article, we discuss how to compute
reduced forms of some symplectic differential systems, arising as variational
equations of hamiltonian systems. We use this to give an effective form of the
Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.
