Meromorphically integrable homogeneous potentials with multiple Darboux points

TL;DR

This paper classifies all meromorphically integrable planar homogeneous potentials with multiple Darboux points, showing that only rotationally invariant potentials qualify, leading to a complete classification of certain real analytic potentials.

Contribution

It proves that the only such integrable potentials with multiple Darboux points are those invariant under rotation, filling a gap in the classification of integrable homogeneous potentials.

Findings

Only rotationally invariant potentials are meromorphically integrable with multiple Darboux points.

Complete classification of integrable real analytic homogeneous potentials of negative degree.

The result addresses a special case in the Maciejewski-Przybylska relation on eigenvalues.

Abstract

We prove that the only meromorphically integrable planar homogeneous potential of degree k <> -2,0,2 having a multiple Darboux point is the potential invariant by rotation. This case is a singular case of the Maciejewski-Przybylska relation on eigenvalues at Darboux points of homogeneous potentials, and needed before a case by case special analysis. The most striking application of this Theorem is the complete classification of integrable real analytic homogeneous potentials in the plane of negative degree.