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We investigate the dynamics near the essential singularity at infinity for a class of zero-free entire vector fields of finite order, i.e., those of the form f(z) = eP(z) where p(z) = Zd or P(z) = a z2 + b z + c. We show that the flow generated by such a vector field has a "bouquet of flowers" attached to the point at infinity.

Dynamics near the essential singularity of a class of entire vector fields