# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Integrability of two dimensional quasi-homogeneous polynomial differential systems. (English) Zbl 1213.37093
The authors deal with polynomial differential systems $$(\dot{x},\dot{y})^{T}=F_{r}=(P,Q)^{T}, \tag1$$ where $F_{r}$ is a quasi-homogeneous polynomial vector field of degree $r\in N \cup {0}$ with respect to type $t=(t_{1},t_{2})\in N^{2},$ i.e., for any arbitrary positive real $\varepsilon$, $P(\varepsilon^{t_{1}}x,\varepsilon^{t_{2}}y)= \varepsilon^{r+t_{1}}P(x,y)$, $Q(\varepsilon^{t_{1}}x,\varepsilon^{t_{2}}y)= \varepsilon^{r+t_{2}}Q(x,y)$ and interested in analyzing when system (1) is analytically integrable.

##### MSC:
 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws
##### Keywords:
polynomial differential systems; Hamiltonian system
Full Text:
##### References:
 [1] A. Algaba, E. Freire, E. Gamero and C. García, The integrability problem for a class of planar systems , Nonlinearity 22 (2009), 395-420. · Zbl 1165.34023 · doi:10.1088/0951-7715/22/2/009 [2] L. Cairó and J. Llibre, Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3, J. Math. Anal. Appl. 331 (2007), 1284-1298. · Zbl 1124.34015 · doi:10.1016/j.jmaa.2006.09.066 [3] J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Local analytic integrability for nilpotent centers , Ergod. Theor. Dynam. Sys. 23 (2003), 417-428. · Zbl 1037.34025 · doi:10.1017/S014338570200127X [4] H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles , Nonlinearity 9 (1996), 501-516. · Zbl 0886.58087 · doi:10.1088/0951-7715/9/2/013 [5] J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems , Nonlinearity 15 (2002), 1269-1280. · Zbl 1024.34001 · doi:10.1088/0951-7715/15/4/313 [6] V.V. Nemytskii and V.V. Stepanov, Qualitative theory of differential equations , Princeton University Press, Princeton, 1960. · Zbl 0089.29502 [7] P.J. Olver, Applications of Lie groups to differential equations , Springer-Verlag, New York, 1986. · Zbl 0588.22001 [8] A. Tsygvintsev, On the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations , J. Phys. A: Math. Gen 34 (2001), 2185-2193. · Zbl 0984.34026 · doi:10.1088/0305-4470/34/11/311