A note on Vishik's normal form

Abstract

The Vishik's Normal Form provides a local smooth conjugation with a linear vector field for smooth vector fields near contacts with a manifold. In the present study, we focus on the analytic case. Our main result ensures that for analytic vector field and manifold, the conjugation with the Vishik's normal form is also analytic. As an application, we investigate the analyticity of Poincar\'e Half Maps defined locally near contacts between analytic vector field and manifold.