Integrable planar homogeneous potentials of degree -1 with small eigenvalues
Annales de l'Institut Fourier, Volume 66 (2016) no. 6, p. 2253-2298

We give a complete classification of meromorphically integrable homogeneous potentials V of degree -1 which are real analytic on 2 {0}. 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 ' (c)=-c,c 1 2 +c 2 2 0 and Sp( 2 V(c)){-1,0,2}. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case V ' (c)=0.

On démontre une classification complète des potentiels V méromorphiquement intégrables homogènes de degré -1, analytiques rééls sur 2 {0}. 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 ' (c)=-c,c 1 2 +c 2 2 0 et Sp( 2 V(c)){-1,0,2}. Enfin, on présente une conjecture pour les autres valeurs propres et le cas des points de Darboux dégénérés V ' (c)=0.

Received : 2013-02-19
Revised : 2014-11-25
Accepted : 2015-01-22
Published online : 2016-10-04
DOI : https://doi.org/10.5802/aif.3063
Classification:  37J30
Keywords: Morales-Ramis theory, homogeneous potentials, D-finiteness, higher variational equations
@article{AIF_2016__66_6_2253_0,
     author = {Combot, Thierry},
     title = {Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {6},
     year = {2016},
     pages = {2253-2298},
     doi = {10.5802/aif.3063},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_6_2253_0}
}
Combot, Thierry. Integrable planar homogeneous potentials of degree $-1$ with small eigenvalues. Annales de l'Institut Fourier, Volume 66 (2016) no. 6, pp. 2253-2298. doi : 10.5802/aif.3063. http://www.numdam.org/item/AIF_2016__66_6_2253_0/

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