A note on the rational non-integrability of the $N$-center problem for almost all degrees of the singularities

In this article, we show that the $N$-center problem with rational weak and moderate forces is not rationally integrable for all but a finite number of values $\alpha \in (0,2)\cap \mathbb{Q}$, where $\alpha$ is the order of the singularities. We identify the remaining cases and provide the necessary conditions for their integrability.