Fifth- and seventh-order nonintegrability conjecture for homogeneous potentials

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 $U'(\theta_0)=0$. For $n\in\mathbb{N}$ with $n\geq 3$, consider the eigenvalue condition

$$\frac{U”(\theta_0)}{U(\theta_0)}-1=\frac{1}{2}(n-1)(n+2).$$

Fifth- and seventh-order nonintegrability conjecture. If $n$ is odd, then the fifth variational equation is not integrable. If $n$ is even and $n\geq 4$, then the seventh variational equation is not integrable.

These assertions sharpen the general conjecture by specifying a fixed obstructing order according to the parity of $n$. They concern higher-order variational equations and the resulting obstructions to integrability of the potential.