Third order integrability conditions for homogeneous potentials of degree -1

TL;DR

This paper establishes a third-order integrability criterion for homogeneous potentials of degree -1 in the plane, applies it to specific polynomial-based potentials, and classifies all meromorphically integrable cases within this family.

Contribution

It introduces a new third-order integrability criterion for degree -1 homogeneous potentials and classifies integrable potentials of a specific form using advanced computational methods.

Findings

Derived a third-order integrability condition for degree -1 potentials

Applied the criterion to polynomial-based potentials of degree less than 3

Classified all meromorphically integrable potentials of the form V(r,θ)=r^{-1}h(e^{iθ})

Abstract

We prove an integrability criterion of order 3 for a homogeneous potential of degree -1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r,\theta)=r^{-1} h(\exp(i\theta)) with h a polynomial of degree less than 3. We find then all meromorphically integrable potentials of this form.