  The problem of averaging for systems with one fast phase was considered from
various points of view in many papers. The averaging method of Krylov and
Bogolyubov and methods of KAM theory originated this line of research, the most
complete results were obtained by Neishtadt, where the coefficients are assumed
real analytic. However, in many problems which are interesting from the point
of view of applications, analytic dependence ceases to hold in the
neighborhoods of some points. We show that one can choose a transformation such
that the averaging procedure of Neishtadt is applicable in these neighborhoods
and such passage to limit is uniform.
