Koenigs embedding flow problem with global C1 smoothness

Which C1 orientation-preserving diffeomorphisms on the compact interval [0,1], that of course fixing the end-points, can be embedded into a C1 flow on [0,1] is a difficult problem in the iteration theory. Such maps were proved to be rare in some references. In this paper this problem is reduced to the relations between solutions of an ordinary differential equation and solutions of a functional equation. By discussing those relations we describe a relatively wide class of orientation-preserving diffeomorphisms on [0,1] which can achieve such a C1 flow. Our results are demonstrated by practical calculation for polygons.