  We describe the equivalence classes of germs of generic 1-parameter families
of complex vector fields z dot = omega_epsilon(z) on C unfolding a singular
point of multiplicity k+1: omega_0 = z^{k+1} + o(z^{k+1}). The equivalence is
under conjugacy by holomorphic change of coordinate and parameter. We provide a
description of the modulus space and (almost) unique normal forms. As a
preparatory step, we present the complete bifurcation diagram of the family of
vector fields z dot = z^{k+1} - epsilon, over CP1.
