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Essential variables in the integrability problem of planar vector fields. (English) Zbl 1241.34018
Summary: This Letter is devoted to the integrability problem of planar nonlinear differential equations. We introduce a new method to detect local analytic integrability or to construct a singular series expansion of the first integral around a singular point for planar vector fields. The method allows to find new variables (essential variables) where the integrability problem is more feasible. The new method can be used in different context and is an alternative to all the methods developed up to now for any particular case.
##### MSC:
 34A34 Nonlinear ODE and systems, general 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 34C05 Location of integral curves, singular points, limit cycles (ODE)