Ordinary differential equations (ODE) correspond to the orientable foliations. Allowing the ODE to be unsolved with respect to the derivative of highest order gives us an interesting class of non-orientable foliations. A method of integration of such differential equations has been suggested by Cayley and Darboux in the context of principal curvature lines on surfaces. Later Hartman and Wintner in the series of works  —  developed a general method of the integration of such ODE’s. In this chapter we introduce the reader to the theory of Hartman and Wintner as well as to the later (geometric) method of A. G. Kuzmin.
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