. Proof and . Obviously, is the direction determined by the Stokes values of generating the collection ( k ); hence, 2 and the set G(( k )) of lemma 2.4 coincides with the set of section 2.2. Thereby, the image of ( k ) is equal to the set of directions determined by the r th roots of the elements of

W. Balser, W. B. Jurkat, and D. A. Lutz, A general theory of invariants for meromorphic di¤erential equations; Part I, formal invariants, Funkcial . Ekvac, vol.22, pp.197-221, 1979.

J. Écalle, Les fonctions résurgentes, tome III : l'équation du pont et la classi?cation analytique des objets locaux, Publ. Math. Orsay, pp.85-90, 1985.

M. Loday-richaud, Rank reduction, normal forms and stokes matrices, Expositiones Mathematicae, vol.19, issue.3, pp.229-250, 2001.
DOI : 10.1016/S0723-0869(01)80003-7

URL : http://doi.org/10.1016/s0723-0869(01)80003-7

M. Loday-richaud and P. Remy, Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems, Journal of Differential Equations, vol.250, issue.3, pp.1591-1630, 2011.
DOI : 10.1016/j.jde.2010.09.018

URL : https://hal.archives-ouvertes.fr/hal-00491614

J. Ramis, Filtration de Gevrey sur le groupe de Picard-Vessiot d'une équation di¤érentielle irrégulière, )). Société Mathématique de France, 1985.

P. Remy, On the Stokes phenomenon of a family of multi-perturbed level-one meromorphic linear di¤erential systems

P. Remy, Matrices de Stokes-Ramis et constantes de connexion pour les syst??mes diff??rentiels lin??aires de niveau unique, Annales de la facult?? des sciences de Toulouse Math??matiques, vol.21, issue.1, pp.93-150, 2012.
DOI : 10.5802/afst.1330

W. Wasow, Asymptotic expansions for ordinary di¤erential equations, 1965.