Geodesic
Quadratic Vector Fields in $\mathbb C\mathrm~P^2$ with Solvable Monodromy Group at Infinity
Informatics and Automation, Nonlinear analytic differential equations, Tome 254 (2006), pp. 130-161 
A. S. Pyartli. Quadratic Vector Fields in $\mathbb C\mathrm~P^2$ with Solvable Monodromy Group at Infinity. Informatics and Automation, Nonlinear analytic differential equations, Tome 254 (2006), pp. 130-161. http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a4/
@article{TRSPY_2006_254_a4,
     author = {A. S. Pyartli},
     title = {Quadratic {Vector} {Fields} in $\mathbb C\mathrm~P^2$ with {Solvable} {Monodromy} {Group} at {Infinity}},
     journal = {Informatics and Automation},
     pages = {130--161},
     year = {2006},
     volume = {254},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a4/}
}
TY  - JOUR
AU  - A. S. Pyartli
TI  - Quadratic Vector Fields in $\mathbb C\mathrm~P^2$ with Solvable Monodromy Group at Infinity
JO  - Informatics and Automation
PY  - 2006
SP  - 130
EP  - 161
VL  - 254
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a4/
LA  - ru
ID  - TRSPY_2006_254_a4
ER  - 
%0 Journal Article
%A A. S. Pyartli
%T Quadratic Vector Fields in $\mathbb C\mathrm~P^2$ with Solvable Monodromy Group at Infinity
%J Informatics and Automation
%D 2006
%P 130-161
%V 254
%U http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a4/
%G ru
%F TRSPY_2006_254_a4

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

Quadratic vector fields for which the line at infinity is a phase curve with three different singular points are considered. It is assumed that the characteristic numbers of these singular points are not multiples of $1/4$ or $1/6$. It is shown that among the fields with fixed characteristic numbers satisfying this assumption, one can choose seven fields such that any other field with solvable noncommutative monodromy group at infinity is affine equivalent to one of the chosen fields. In addition, quadratic vector fields with commutative monodromy group at infinity are described.

[1] Ilyashenko Yu.S., Pyartli A.S., “The monodromy group at infinity of a generic polynomial vector field on the complex projective plane”, Russ. J. Math. Phys., 2:3 (1994), 275–315 | MR | Zbl

[2] Ilyashenko Yu.S., Pyartli A.S., “Ratsionalnye differentsialnye uravneniya s nesvobodnoi gruppoi monodromii na beskonechnosti”, Tr. MIAN, 213, 1997, 56–73 | MR | Zbl

[3] Pyartli A.S., “Ratsionalnye differentsialnye uravneniya s kommutativnoi gruppoi monodromii na beskonechnosti”, Tr. Mosk. mat. o-va, 61, 2000, 75–106 | MR | Zbl

[4] Ortiz-Bobadilla L., “Quadratic vector fields in $\mathbb C\mathrm P^2$ with two suddle–node type singularities at infinity”, J. Dyn. and Control Syst., 1:3 (1995), 295–317 | DOI | MR | Zbl

[5] Ortis-Bobadilya L., “Nelineinoe yavlenie Stoksa v chetyrekhparametricheskom semeistve polinomialnykh differentsialnykh uravnenii poryadka vyshe dvukh s osobennostyami tipa sedlouzel na beskonechnosti”, Tr. MIAN, 213, 1997, 74–89 | MR | Zbl

[6] Ilyashenko Yu., “Centennial history of Hilbert's 16th problem”, Bull. Amer. Math. Soc., 39:3 (2002), 301–354 | DOI | MR | Zbl

[7] Elizarov P.M., Ilyashenko Yu.S., Shcherbakov A.A., Voronin S.M., “Finitely generated groups of germs of one-dimensional conformal mappings, and invariants for complex singular points of analytic foliations of the complex plane”, Adv. Sov. Math., 14 (1993), 57–105 | MR | Zbl

[8] Scherbakov A.A., “Dinamika lokalnykh grupp konformnykh otobrazhenii i obschie svoistva differentsialnykh uravnenii na $\mathbb C^2$”, Tr. MIAN, 254 (2006), 111–129