  Through blowup \cite{T}, the rescaled Poicar\'e map is continued to
nondegenerate singularities in \cite{GY,CY}. For a nondegenerate singularity of
a $C^1$ vector field, we prove that the extended rescaled Poincar\'e map over
the singurity equals the counterpart of the vector field's linearization. For
singularities of two dimensional linear vector fields of three types, we
compute the rescaled Poincar\'e maps upto the second order. We show that except
for the focus case, the second order generally does not vanish, and the
rescaled Poincar\'e map is generally nonlinear.
