  Considering systems of separations in a graph that separate every pair of a
given set of vertex sets that are themselves not separated by these
separations, we determine conditions under which such a separation system
contains a nested subsystem that still separates those sets and is invariant
under the automorphisms of the graph.
  As an application, we show that the $k$-blocks -- the maximal vertex sets
that cannot be separated by at most $k$ vertices -- of a graph $G$ live in
distinct parts of a suitable tree-decomposition of $G$ of adhesion at most $k$,
whose decomposition tree is invariant under the automorphisms of $G$. This
extends recent work of Dunwoody and Kr\"on and, like theirs, generalizes a
similar theorem of Tutte for $k=2$.
  Under mild additional assumptions, which are necessary, our decompositions
can be combined into one overall tree-decomposition that distinguishes, for all
$k$ simultaneously, all the $k$-blocks of a finite graph.
