Generic complex vector fields in R2

In this paper we study what kind of general complex vector fields in R2 one is most likely to encounter and study their singularities. We first construct a dense and open subset of nonvanishing complex vector fields under certain topology, whose elements are called generic complex vector fields. Then we show that a generic complex vector field can be reduced to either the Cauchy–Riemann operator or a Mizohata type operator or a cuspidal operator locally. Finally we note that an integral of any cuspidal operator is of the form (xy+x3,y) locally after coordinates changes.