  The KdV hierarchy is a family of evolutions on a Schr\"odinger operator that
preserves its spectrum. Canonical systems are a generalization of Schr\"odinger
operators, that nevertheless share many features with Schr\"odinger operators.
Since this is a very natural generalization, one would expect that it would
also be straightforward to build a hierarchy of isospectral evolutions on
canonical systems analogous to the KdV hierarchy. Surprisingly, we show that
there are many obstructions to constructing a hierarchy of flows on canonical
systems that obeys the standard assumptions of the KdV hierarchy. This suggests
that we need a more sophisticated approach to develop such a hierarchy, if it
is indeed possible to do so.
