Chains of integrally closed ideals

Let $(A,\mathfrak{m})$ be an excellent normal local ring with algebraically closed residue class field. Given integrally closed $\mathfrak{m}$-primary ideals $I\supset J$,weshowthatthereisacompositionseriesbetween$I$and$J$, by integrally closed ideals only. Also we show that any given integrally closed $\text{fm}$-primary ideal $I$, the family of integrally closed ideals $J\subset I, l_A(I/J)=1$formsanalgebraicvarietywithdimension$\dim A -1$.