Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues

Combot, Thierry

Annales de l'Institut Fourier, Tome 66 (2016) no. 6, pp. 2253-2298.

On démontre une classification complète des potentiels $V$ méromorphiquement intégrables homogènes de degré $-1$, analytiques rééls sur ${ℝ}^2\setminus \left\{0\right\}$. Dans le cas plus général où $V$ est seulement méromorphe sur un ouvert d’une variété algébrique, on démontre une classification de tous les potentiels intégrables ayant un point de Darboux $c$ tel que $V^{\text{'}}\left(c\right)=-c,c_1^2+c_2^2\ne 0$ et $\text{Sp}\left({\nabla }^{2}V\left(c\right)\right)\subset \{-1,0,2\}$. Enfin, on présente une conjecture pour les autres valeurs propres et le cas des points de Darboux dégénérés $V^{\text{'}}\left(c\right)=0$.

We give a complete classification of meromorphically integrable homogeneous potentials $V$ of degree $-1$ which are real analytic on ${ℝ}^2\setminus \left\{0\right\}$. In the more general case when $V$ is only meromorphic on an open set of an algebraic variety, we give a classification of all integrable potentials having a Darboux point $c$ with $V^{\text{'}}\left(c\right)=-c,c_1^2+c_2^2\ne 0$ and $\text{Sp}\left({\nabla }^{2}V\left(c\right)\right)\subset \{-1,0,2\}$. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case $V^{\text{'}}\left(c\right)=0$.

Classification : 37J30
Keywords: Morales-Ramis theory, homogeneous potentials, D-finiteness, higher variational equations
Mot clés : Théorie de Morales-Ramis, potentiels homogènes, D-finitude, équations variationelles supérieures

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