Qualitative classification of singular points

We study in this paper the qualitative classification of isolated singular points of analytic differential equations in the plane. Two singular points are said to be qualitatively equivalent if they are topologically equivalent and furthermore, two orbits start or end in the same direction at one singular point if and only if the equivalent two orbits start or end in the same direction at the other singular point. The degree of the leading terms in the Taylor expansion of a differential equation at a singular point will be called the degree of this singular point. The qualitative equivalence divides the set of singular points of degreem, into equivalence classes. The main problems studied here are the characterization of all qualitative equivalence classes and then, to determine to which class a singular point will belong to. We remark that up to now these problems have been solved only in the casem=1 before.