Nonintegrability conjecture for homogeneous potentials with small eigenvalues

Let $V=\frac{1}{r}U(\theta)$ be a homogeneous potential of degree $-1$ in the plane, in polar coordinates, with $U$ meromorphic and $2\pi$-periodic. Suppose that there exists $\theta_0$ such that $U'(\theta_0)=0$ and

$$\frac{U”(\theta_0)}{U(\theta_0)}-1=\frac{1}{2}(n-1)(n+2),\qquad n\in\mathbb{N},\quad n\geq 3.$$

Nonintegrability conjecture. There exists $k\in\mathbb{N}^*$ such that the variational equation at order $k$ is not integrable.

This predicts an obstruction to meromorphic integrability for homogeneous potentials whose Darboux-point eigenvalue belongs to the specified discrete family. The conjecture does not identify the obstructing order in general; the paper gives more specific nonintegrability results for odd and even values of $n$.