  Eigenstate thermalization is widely accepted as the mechanism behind
thermalization in generic isolated quantum systems. Using the example of a
single magnetic defect embedded in the integrable spin-1/2 $XXZ$ chain, we show
that locally perturbing an integrable system can give rise to eigenstate
thermalization. Unique to such setups is the fact that thermodynamic and
transport properties of the unperturbed integrable chain emerge in properties
of the eigenstates of the perturbed (nonintegrable) one. Specifically, we show
that the diagonal matrix elements of observables in the perturbed eigenstates
follow the microcanonical predictions for the integrable model, and that the
ballistic character of spin transport in the integrable model is manifest in
the behavior of the off-diagonal matrix elements of the current operator in the
perturbed eigenstates.
