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

Abstract

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