Integrability of potentials of degree $k \neq \pm 2$. Second order variational equations between Kolchin solvability and Abelianity

Abstract: In our previous paper: Integrability of Homogeneous potentials of degree $k = \pm 2$. An application of higher order variational equations, we tried to extract some particular structures of the higher variational equations (the ${VE}_p$ for $p >1 $), along particular solutions of some Hamiltonian systems. Then, we use them to get new Galois obstructions to the integrability of natural Hamiltonian with potential of degree $k = \pm 2$. In the present work, we apply the results of the previous paper, to the complementary cases, when the degrees of the potentials are relative integers $k$, with $|k| >2$. Since these cases are much more general and complicated, we reduce our study only to the second variational equation ${VE}_2$.