Product of distinct simple integrally closed ideals in 2-dimensional regular local rings

Let $(R,m)$ be a two-dimensional regular local ring and $I$ an $m$-primary integrally closed ideal in $R$. In this paper, we give equivalent conditions for $I$ to be a product of distinct simple $m$-primary integrally closed ideals (i.e., $I = I_{1}\cdots I_{l}$, where $I_{1},\cdots ,I_{l}$ are distinct simple $m$-primary integrally closed ideals of $R$) in terms of the regularity of $R[It]/p$ for all $p \in \text {Min} (mR[It])$ and in terms of how to choose a minimal generating set for $I$ over its minimal reductions.