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In this paper we describe the bifurcation diagram of the$2$-parameter familyof vector fields $\dot z = z(z^k+\epsilon_1z+\epsilon_0)$ over $\mathbbC\mathbb P^1$ for $(\epsilon_1,\epsilon_0)\in \mathbb C^2$. There are two kindsof bifurcations: bifurcations of parabolic points and bifurcations ofhomoclinic loops through infinity. The latter are studied using the tool of theperiodgon introduced in a particular case in \cite{CR}, and then generalized in\cite{KR}. We apply the results to the bifurcation diagram of a generic germ of2-parameter analytic unfolding preserving the origin of the vector field $\dotz = z^{k+1} +o(z^{k+1})$ with a parabolic point at the origin.

Two-parameter unfolding of a parabolic point of a vector field in $\mathbb C$ fixing the origin