A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems

Let $[A]: Y=AY$ with $Ain mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $Rin {cal M}_{n}(bar{k})$ is a {em reduced form} of $[A]$ if $Rin mathfrak{g}(bar{k})$ and there exists $Pin GL_n (bar{k})$ 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.