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Article Dans Une Revue Annales de l'Institut Fourier Année : 2016

Integrable planar homogeneous potentials of degree −1 with small eigenvalues

Résumé

We give a complete classification of meromorphically integrable homogeneous potentials $V$ of degree $-1$ which are real analytic on $\mathbb{R}^2\setminus\{0\}$. 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 $V'(c)=-c,\; c_1^2+c_2^2\neq 0$ and $\hbox{Sp}(\nabla^2 V(c)) \subset\{-1,0,2\}$. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case $V'(c)=0$.

Dates et versions

hal-01493049 , version 1 (20-03-2017)

Identifiants

Citer

Thierry Combot. Integrable planar homogeneous potentials of degree −1 with small eigenvalues. Annales de l'Institut Fourier, 2016, 66 (6), pp.2253-2298 ⟨10.5802/aif.3063⟩. ⟨hal-01493049⟩
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