  If $f$ is an endomorphism of a finite dimensional vector space over a field
$K$ then an invariant subspace $X \subseteq V$ is called hyperinvariant
(respectively, characteristic) if $X$ is invariant under all endomorphisms
(respectively, automorphisms) that commute with $f$. According to Shoda (Math.
Zeit. 31, 611--624, 1930) only if $|K| = 2$ then there exist endomorphisms $f$
with invariant subspaces that are characteristic but not hyperinvariant. In
this paper we obtain a description of the set of all characteristic
non-hyperinvariant subspaces for nilpotent maps $f$ with exactly two unrepeated
elementary divisors.
