@article{10272/25455,
year = {2012},
month = {10},
url = {https://hdl.handle.net/10272/25455},
abstract = {We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systems ? = (?h/?y (x,y)K (h,y n) + y n-1 ?(h,y n) ?(x,y), ? = -?h/?x(x,y)K(h,y n) ?(x,y), where h, K, ? and ? are analytic functions defined in the neighbourhood of O with K(O) ? 0 or ?(O) ? 0 and n ? 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know},
organization = {This work has been partially supported by  Ministerio de
Ciencia e Innovación, co-financed with
FEDER funds, in the frame of the project MTM2010-20907-C02-02 and by 
Consejería de Educación y Ciencia de la Junta de
Andalucía (FQM-276 and EXC/2008/FQM-872)},
publisher = {Cambridge University Press},
keywords = {Integrabilidad de sistemas diferenciales},
title = {A note on analytic integrability of planar vector fields},
author = {Algaba Durán, Antonio and García García, Cristóbal and Reyes Columé, Manuel},
}