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Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations

The present work is the first of a serie of two papers, in which we analyse the higher variational equations associated to natural Hamiltonian systems, in their attempt to give Galois obstruction to their integrability. We show that the higher variational equations $VE_{p}$ for $p\geq 2$, although complicated they are, have very particular algebraic structure. Preceisely they are solvable if $VE_{1}$ is virtually Abelian since they are solvable inductively by what we call the \emph{second level integrals}. We then give necessary and sufficient conditions in terms of these second level integrals for $VE_{p}$ to be virtually Abelian (see Theorem 3.1). Then, we apply the above to potentials of degree $k=\pm 2$ by considering their $VE_{p}$ along Darboux points. And this because their $VE_{1}$ does not give any obstruction to the integrablity. In Theorem 1.2, we show that under non-resonance conditions, the only degree two integrable potential is the \emph{harmonic oscillator}. In contrast for degree -2 potentials, all the $VE_{p}$ along Darboux points are virtually Abelian (see Theorem 1.3)