Peixoto graphs of Morse-Smale foliations on surfaces

We consider non-orientable foliations, given on orientable compact 2-manifolds. Within a special class of such foliations, a Morse-Smale class, a complete topological classification is proposed. Our main result is a generalization of the well-known M.M. Peixotoʹs theorems (Peixoto, 1973), to the case of foliations on surfaces. In a whole, the paper gives an understanding of the ‘topological information’ contained in a Morse-Smale non-orientable foliation.