  This work is a continuation of [Y. Fittouhi and A. Joseph, Weierstrass
Sections for Parabolic adjoint action in type $A$]. Let $G$ be an irreducible
simple algebraic group and $B$ a Borel subgroup of $G$. Let $\mathfrak n$ be
the Lie algebra of the nilradical of $B$. Consider an irreducible subgroup $P$
of $G$ containing $B$. Let $P'$ be the derived group of $P$. Let $\mathfrak m$
be the Lie algebra of the nilradical of $P$. A theorem of Richardson asserts
that the algebra $\mathbb C[\mathfrak m]^{P'}$ of $P$ semi-invariants is
multiplicity-free. A linear subvariety $e+V$ such that the restriction map
induces an isomorphism of $\mathbb C[\mathfrak m]^{P'}$ onto $\mathbb C[e+V]$
is called a Weierstrass section for the action of $P'$ on $\mathfrak m$. Here
in type $A$ such a section is constructed, but in better form than that given
in Sect. 4, loc cit. Yet the main difference is a complete change of emphasis
from the construction of a Weierstrass section, to its application.
  Let $\mathscr N$ be the nilfibre relative to this action. From the
construction of a Weierstrass section $e+V$, it is shown that $e \in \mathscr
N$. Then $P.e$ is contained in a unique irreducible component $\mathscr C$ of
$\mathscr N$. The structure of $e+V$ is used to give a rather explicit
description of $\mathscr C$ as a $B$ saturation set, that is of the form
$\overline{B.\mathfrak u}$, where $\mathfrak u$ is a subalgebra of $\mathfrak
n$ . This algebra is not necessarily complemented by a subalgebra in $\mathfrak
n$ and so $\overline{B.\mathfrak u}$ is not necessarily an orbital variety
closure (hence Lagrangian) but it can be.
  It is shown that $\mathscr C$ need not contain a dense $P$ orbit and this by
a purely theoretical analysis. This occurs for an appropriate parabolic in
$A_{10}$ and is possibly the simplest example. In this particular case
$\mathscr C$ is not an orbital variety closure.
