This talk will lie at the crossroads of Differential Galois Theory and Hamiltonian Systems.
 
We define a notion of "reduced form" for a  linear differential system $Y'=AY\, \quad A\in M_{n}(k)$ as a gauge equivalent differential system $Z'=BZ$ such that $B\in\mathfrak{g}(\bar{k})$, where $k$ is a $C_1$ differential field and  $\mathfrak{g}:=Lie(Y'=AY)$.   Intuitively, a reduced form is the sparsest possible gauge equivalent form for a differential linear system. Such a form is very useful because it is considerably simpler to solve a system which has been put under reduced form than a system that has not.
 
A Hamiltonian System is (Liouville completely) integrable if it possesses as many independent first integrals in pairwise involution as degrees of freedom. The Morales Ramis Theorem tells us that if a Hamiltonian System is integrable then the Lie algebra of its variational equation (along a particular solution different from an equilibrium point) will be abelian.
 
An algorithm is shown such that, given a linear Hamiltonian system $Y'=AY$ with $A\in \mathfrak{sp}(4,k)$ , puts $A$ into reduced form whenever $Lie(Y'=AY)$ is abelian.  Otherwise, the algorithm tells us that $Lie(Y'=AY)$ is not abelian. 
If we apply this algorithm in the context of Morales-Ramis' Theorem, we obtain an algorithm that tells us either  a Hamiltonian system with 2 degrees of freedom is not abelian or returns a reduced form of its Variational Equation. (Application : the Hill system is not integrable.  Our proof is faster than the original one given by Morales-Simo-Simon).

If  $Lie(VE)$ is abelian our algorithm allows us to reduce $(VE)$ and almost in the same stride all higher order variational equations. 
One interest of higher variational equations is that they provide the propper context for th application of Morales-Ramis-Simó's Theorem. The other, on wich we focus in this talk, is the re-construction of limited Taylor developments of  formal first integrals. We will give some hints on this matter.

This is a joint work with Sergi Simon (Portsmouth Univ.) and Jacques-Arthur Weil (Univ. Limoges).