English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

— We give a complete classification of meromorphically integrable homogeneous potentials V of degree −1 which are real analytic on R2 \ {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, c1 + c2 6= 0 and Sp(∇2V (c)) ⊂ {−1, 0, 2}. We eventually present a conjecture for the other eigenvalues and the degenerate Darboux point case V ′(c) = 0. Résumé. — On démontre une classification complète des potentiels V méromorphiquement intégrables homogènes de degré −1, analytiques rééls sur R2 \{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, c1 +c2 6= 0 et Sp(∇2V (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. The problem of finding potentials which are integrable in the Liouville sense is a difficult and ancient problem. Liouville found that finding enough first integrals (n for a n-dimensional potential) allows to solve the differential system associated to the potential by quadrature (the potential is then called integrable). The main difficulty is to find these first integrals, as they do not always exists, at least not globally. Almost all integrable rational potentials have rather simple first integrals, but one cannot even exclude very high degree rational first integrals. So one of the main problem is to find all integrable potentials, and certify that no others exist. A Theorem from Morales-Ramis-Simo (Theorem 2 in [25]) gives necessary conditions for integrability with meromorphic first integrals (see also earlier versions of this Theorem in [31, 21, 22]). The differential Galois

Integrable planar homogeneous potentials of degree -1 with small eigenvalues