The analytic and formal normal form for the nilpotent singularity. The case of generalized saddle-node (Q1865010)

The generalized saddle-node case of certain vector fields is considered. The aim is to provide analogous results to those of the same author and \textit{H. Żolądek} [J. Differ. Equations 179, 479--537 (2002; Zbl 1005.34034)], and of other authors cited in the current paper, in the generalized cusp case. It is shown that two germs of analytic vector fields (with generalized saddle-node singularity) are analytically equivalent if and only if their monodromy groups are analytically equivalent. The proof makes use of a restricted version of the same theorem due to \textit{M. Berthier, R. Meziani} and \textit{P. Sad} [Bull. Sci. Math. 123, 351--370 (1999; Zbl 0963.32021)].