  Let $k$ be a field. We consider triples $(V,U,T)$, where $V$ is a finite
dimensional $k$-space, $U$ a subspace of $V$ and $T \:V \to V$ a linear
operator with $T^n = 0$ for some $n$, and such that $T(U) \subseteq U$. Thus,
$T$ is a nilpotent operator on $V$, and $U$ is an invariant subspace with
respect to $T$.
  We will discuss the question whether it is possible to classify these
triples. These triples $(V,U,T)$ are the objects of a category with the
Krull-Remak-Schmidt property, thus it will be sufficient to deal with
indecomposable triples. Obviously, the classification problem depends on $n$,
and it will turn out that the decisive case is $n=6.$ For $n < 6$, there are
only finitely many isomorphism classes of indecomposables triples, whereas for
$n > 6$ we deal with what is called ``wild'' representation type, so no
complete classification can be expected.
  For $n=6$, we will exhibit a complete description of all the indecomposable
triples.
