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DCDS

The present work is the first one of two papers, in which we analyse
systems of higher order variational equations associated to natural
Hamiltonian systems with homogeneous potential of degree
$k\in\mathbb{Z}\setminus \{-1,0,1\}$. Our attempt is to give
necessary conditions for complete integrability which can be deduced
in a framework of differential Galois theory. We show that the
higher variational equations $\mathrm{VE}_p$ of order $p\geq 2$,
although complicated, have a very particular algebraic structure.
More precisely, we show that if $\mathrm{VE}_1$ has virtually
Abelian differential Galois group (DGG), then $\mathrm{VE}_{p}$
are solvable for an arbitrary $p>1$. We proved this inductively using
what we call the

*second level integrals*. Then we formulate the necessary and sufficient conditions in terms of these second level integrals for $\mathrm{VE}_{p}$ to be virtually Abelian. We apply the above conditions to potentials of degree $k=\pm 2$ considering their $\mathrm{VE}_p$ with $p>1$ along Darboux points. For $k= 2$, $\mathrm{VE}_1$ does not give any obstruction to the integrability. We show that under certain non-resonance condition, the only degree two integrable potential is the*multidimensional harmonic oscillator*. In contrast, for degree $k=-2$ potentials, all the $\mathrm{VE}_{p}$ along Darboux points are virtually Abelian.
DCDS

In our previous paper [4], we tried to extract some particular
structures of the higher variational equations (the $\mathrm{VE}_p$ for $p
\geq 2$), along particular solutions of natural Hamiltonian systems with
homogeneous potential of degree $k=\pm 2$. We investigate these variational
equations in a framework of differential Galois theory. Our aim was to obtain
new obstructions for complete integrability. In this paper we extend
results of [4] to the complementary cases, when the homogeneous
potential has integer degree of homogeneity $k\in\mathbb{Z}$, and $|k| \geq 3$.
Since these cases are much more general and complicated, we restrict our
study only to the second order variational equation $\mathrm{VE}_2$.

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