Geodesic
Nonintegrabilty of a Halphen system
Teoretičeskaâ i matematičeskaâ fizika, Tome 181 (2014) no. 2, pp. 296-311

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the Halphen system with real variables and real constants. We show that in the case where at least one constant is nonzero, this system does not admit any first integral that can be described by formal power series. It hence follows that analytic first integrals do not exist. Furthermore, we prove that first integrals of the Darboux type also do not exist.
Keywords: Halphen system, analytic first integral. 
J. Llibre; C. Valls. Nonintegrabilty of a Halphen system. Teoretičeskaâ i matematičeskaâ fizika, Tome 181 (2014) no. 2, pp. 296-311. http://geodesic.mathdoc.fr/item/TMF_2014_181_2_a3/
@article{TMF_2014_181_2_a3,
     author = {J. Llibre and C. Valls},
     title = {Nonintegrabilty of {a~Halphen} system},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {296--311},
     year = {2014},
     volume = {181},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2014_181_2_a3/}
}
TY  - JOUR
AU  - J. Llibre
AU  - C. Valls
TI  - Nonintegrabilty of a Halphen system
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2014
SP  - 296
EP  - 311
VL  - 181
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2014_181_2_a3/
LA  - ru
ID  - TMF_2014_181_2_a3
ER  - 
%0 Journal Article
%A J. Llibre
%A C. Valls
%T Nonintegrabilty of a Halphen system
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2014
%P 296-311
%V 181
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2014_181_2_a3/
%G ru
%F TMF_2014_181_2_a3

[1] G. Darboux, Ann. Sci. École Norm. Sup. (2), 7 (1878), 101–150 | DOI | MR

[2] G. Gibbons, C. Pope, Commun. Math. Phys., 66:3 (1979), 267–290 | DOI | MR

[3] G. Halphen, C. R. Acad. Sci. Paris, 92 (1881), 1101–1103

[4] M. Atya, N. Khitchin, Geometriya i dinamika magnitnykh monopolei, Mir, M., 1991 | MR

[5] B. Dubrovin, “Geometry of the $2$D topological field theories”, Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[6] I. A. Strachan, TMF, 99:3 (1994), 545–551 | MR | Zbl

[7] L. A. Takhtadzhyan, TMF, 93:2 (1992), 330–341 | MR | Zbl

[8] L. Takhtajan, Commun. Math. Phys., 160 (1994), 295–315 | DOI | MR | Zbl

[9] A. Maciejewski, J. Strelcyn, Phys. Lett. A, 201 (1995), 161–166 | DOI | MR | Zbl

[10] C. Valls, J. Geom. Phys., 56 (2006), 1192–1197 | DOI | MR | Zbl

[11] C. Valls, J. Geom. Phys., 57 (2006), 89–100 | DOI | MR | Zbl

[12] J. Giné, J. Llibre, Pacific J. Math., 218 (2005), 343–355 | DOI | MR | Zbl

[13] F. Dumortier, J. Llibre, J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006 | MR | Zbl

[14] J. Llibre, X. Zhang, Nonlinearity, 15 (2002), 1269–1280 | DOI | MR | Zbl

[15] J. Llibre, C. Valls, J. Math. Phys., 46 (2005), 072901, 13 pp. | DOI | MR | Zbl

[16] J. Llibre, X. Zhang, J. Differential Equations, 246 (2009), 541–551 | DOI | MR | Zbl

[17] J. Llibre, X. Zhang, Bull. Sci. Math., 133 (2009), 765–778 | DOI | MR | Zbl