Geodesic
On integrability of a planar system of ODEs near a degenerate stationary point
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 34-47 
A. D. Bruno; V. F. Edneral. On integrability of a planar system of ODEs near a degenerate stationary point. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 34-47. http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a1/
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     author = {A. D. Bruno and V. F. Edneral},
     title = {On integrability of a~planar system of {ODEs} near a~degenerate stationary point},
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     year = {2009},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a1/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We consider an autonomous system of ordinary differential equations, which is resolved with respect to the derivatives. To study local integrability of the system near a degenerate stationary point, we use an approach based on Power Geometry and on the computation of the resonant normal form. For the concrete planar 5-parameter system, we found the complete set of necessary conditions on parameters of the system for which the system is locally integrable near a degenerate stationary point. This set consists of 4 two-parameter sets in this 5-parameter space. For 3 such sets we found sufficient conditions of a local integrability by independent methods. Because these methods are constructive we get first integrals of the system. So at these set of parameters, the system is globally integrable for these 3 sets. For the forth set we have at the moment only approximations of the local integrals as truncated power series in parameters of the system, but we believe that it is possible to sum them up to finite functions. Bibl. – 8 titles.

[1] A. D. Bryuno, V. F. Edneral, “Algoritmicheskii analiz lokalnoi integriruemosti”, DAN, 424:3 (2009), 299–303 | MR

[2] A. D. Bryuno, Stepennaya geometriya v algebraicheskikh i differentsialnykh uravneniyakh, Fizmatlit, M., 1998 | MR | Zbl

[3] A. D. Bryuno, Lokalnyi metod nelineinogo analiza differentsialnykh uravnenii, Nauka, M., 1979 | MR

[4] A. D. Bryuno, “Analiticheskaya forma differentsialnykh uravnenii”, Trudy moskovskogo matematicheskogo obschestva, 25, 1971, 119–262 ; ibid, 26, 1972, 199–239 | Zbl | Zbl

[5] A. Algaba, E. Gamero, C. Garcia, “The integrability problem for a class of planar systems”, Nonlinearity, 22 (2009), 395–420 | DOI | MR | Zbl

[6] V. F. Edneral, “On algorithm of the normal form building”, Proceedings of the CASC 2007, LNCS, 4770, eds. Ganzha et al., Springer-Verlag, 2007, 134–142 | Zbl

[7] K. L. Zigel, Lektsii o nebesnoi mekhanike, Fizmatlit, M., 1959

[8] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, Tom 1, Nauka, M., 1973