We give an equivalent definition of the local volume of an isolated singularity $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\rm Vol}_{\text {BdFF}}(X,0)$ given in [S. Boucksom, T. de Fernex, C. Favre, The volume of an isolated singularity. Duke Math. J. 161 (2012), 1455–1520] in the $\mathbb{Q}$-Gorenstein case and we generalize it to the non-$\mathbb{Q}$-Gorenstein case. We prove that there is a positive lower bound depending only on the dimension for the non-zero local volume of an isolated singularity if $X$ is Gorenstein. We also give a non-$\mathbb{Q}$-Gorenstein example with ${\rm Vol}_{\text {BdFF}}(X,0)=0$, which does not allow a boundary $\Delta $ such that the pair $(X,\Delta )$ is log canonical.

In this paper we solve the problem of desingularization of an absolutely isolated singularity of a differential equation, including the dicritical case. As an application, we prove the finiteness of the number of dicritical points in the blowing up tree of an absolutely isolated singularity.