  One dimensional systems sometimes show pathologically slow decay of currents.
This robustness can be traced to the fact that an integrable model is nearby in
parameter space. In integrable models some part of the current can be
conserved, explaining this slow decay. Unfortunately, although this
conservation law is formally anticipated, in practice it has been difficult to
find in concrete cases, such as the Heisenberg model. We investigate this issue
both analytically and numerically and find that the appropriate conservation
law can be a non-analytic combination of the known local conservation laws and
hence is invisible to elementary assumptions.
