# 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)
The integrability problem for a class of planar systems. (English) Zbl 1165.34023

The paper is a study of the integrability problem for the planar systems $\stackrel{˙}{𝐱}=𝐅\left(𝐱\right)$, where $𝐱=\left(x,y\right)\in {ℝ}^{2}$, and $𝐅\left(𝐱\right)$ is a formal or analytic vector-function that vanishes at the origin and can be expressed as a perturbation of a generic Hamiltonian quasi-homogeneous vector-function with higher-degree quasi-homogeneous terms (such type of functions is naturally connected with Newton polyhedra and often used in the analysis of singularities. The authors consider the formal integrability problem for such systems, but in some analytical cases the convergence of the corresponding formal series can be established, and the formal integrability implies the analytical one.

The main result of the paper is that the quoted system with some additional conditions is formally integrable if and only if it is formally conjugated, via dissipative transformations, to a divergence-free system. To derive this result, the authors obtain a suitable quasi-homogeneous ${C}^{\infty }$-smooth normal form for such systems. As an application they compute the above quasi-homogeneous normal form for various concrete systems, for instance, for the Bogdanov–Takens singularities of different types (nilpotent linear part) and for a family of planar systems with zero linear part, and obtain the explicit conditions of the formal integrability for these systems.

##### MSC:
 34C20 Transformation and reduction of ODE and systems, normal forms 34C05 Location of integral curves, singular points, limit cycles (ODE) 34A05 Methods of solution of ODE