Morales–Ramis-type conjecture for Newton systems at Darboux points

Let a Newton system of the form referred to in the source be meromorphically integrable in the Jacobi sense in a neighborhood of a phase curve $\Gamma$ corresponding to a Darboux point. For each $k\in\mathbb{N}$, consider its $k$-th order variational equations and their differential Galois group.

Differential Galois integrability conjecture. The identity component of the differential Galois group of the $k$-th order variational equations is Abelian for every $k\in\mathbb{N}$.

This is proposed as an extension of the corresponding theorem for Hamiltonian systems to Newton equations. The paper explains that higher-order differential Galois groups are difficult to analyse and that extending the Hamiltonian result to Newton systems presents theoretical problems, so the assertion remains open.