  Let $\mathbf{k}$ be a differential field and let $[A]\,:\,Y'=A\,Y$ be a
linear differential system where $A\in\mathrm{Mat}(n\,,\,\mathbf{k})$. We say
that $A$ is in a reduced form if $A\in\mathfrak{g}(\bar{\mathbf{k}})$ where
$\mathfrak{g}$ is the Lie algebra of $[A]$ and $\bar{\mathbf{k}}$ denotes the
algebraic closure of $\mathbf{k}$. We owe the existence of such reduced forms
to a result due to Kolchin and Kovacic \cite{Ko71a}. This paper is devoted to
the study of reduced forms, of (higher order) variational equations along a
particular solution of a complex analytical hamiltonian system $X$. Using a
previous result \cite{ApWea}, we will assume that the first order variational
equation has an abelian Lie algebra so that, at first order, there are no
Galoisian obstructions to Liouville integrability. We give a strategy to
(partially) reduce the variational equations at order $m+1$ if the variational
equations at order $m$ are already in a reduced form and their Lie algebra is
abelian. Our procedure stops when we meet obstructions to the meromorphic
integrability of $X$. We make strong use both of the lower block triangular
structure of the variational equations and of the notion of associated Lie
algebra of a linear differential system (based on the works of Wei and Norman
in \cite{WeNo63a}). Obstructions to integrability appear when at some step we
obtain a non-trivial commutator between a diagonal element and a nilpotent
(subdiagonal) element of the associated Lie algebra. We use our method coupled
with a reasoning on polylogarithms to give a new and systematic proof of the
non-integrability of the H\'enon-Heiles system. We conjecture that our method
is not only a partial reduction procedure but a complete reduction algorithm.
In the context of complex Hamiltonian systems, this would mean that our method
would be an effective version of the Morales-Ramis-Sim\'o theorem.
