Let $\cal O$ be a nilpotent orbit in the Lie algebra ${ s}{ l}_n({\Bbb C})$ and let $\cal V$ be an orbital variety contained in $\cal O$. Let $\bf P$ be the largest parabolic subgroup of $SL (n,{\Bbb C})$ stabilizing $\cal V$. We describe nilpotent orbits such that all the orbital varieties in them have a dense $\bf P$ orbit and show that for $n$ big enough the majority of nilpotent orbits do not fulfill this.