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Meromorphic infinitesimal affine actions of the plane

We study complex Lie algebras spanned by pairs \left(Z,Y\right) of germs of a meromorphic vector field of the complex plane satisfying \left[Z,Y\right]=δY for some δ\in\ww C . This topic relates to Liouville-integrability of the differential equation induced by the foliation underlying Z . We give a direct geometric proof of a result by M. Berthier and F. Touzet characterizing germs of a foliation admitting a first-integral in a Liouvillian extension of the standard differential field. In so doing we study transverse and tangential rigidity properties when Z is holomorphic and its linear part is not nilpotent. A second part of the article is devoted to computing the Galois-Malgrange groupoid of meromorphic Z .