  An improved PV-reduction method for one-loop integrals with auxiliary vector
$R$ has been proposed in \cite{Feng:2021enk,Hu:2021nia}. It has also been shown
that the new method is a self-completed method in \cite{Feng:2022uqp}. Analytic
reduction coefficients can be easily produced by recursion relations in this
method, where the Gram determinant appears in denominators. The singularity
caused by Gram determinant is a well-known fact and it is important to address
these divergences in a given frame. In this paper, we propose a systematical
algorithm to deal with this problem in our method. The key idea is that now the
master integral of the highest topology will be decomposed into combinations of
master integrals of lower topologies. By demanding the cancellation of
divergence for obtained general reduction coefficients, we solve decomposition
coefficients as a Taylor series of the Gram determinant. Moreover, the same
idea can be applied to other kinds of divergences.
