  We describe hierarchies of exact string backgrounds obtained as non-Abelian
cosets of orthogonal groups and having a space--time realization in terms of
gauged WZW models. For each member in these hierarchies, the target-space
backgrounds are generated by the ``boundary'' backgrounds of the next member.
We explicitly demonstrate that this property holds to all orders in $\alpha'$.
It is a consequence of the existence of an integrable marginal operator build
on, generically, non-Abelian parafermion bilinears. These are dressed with the
dilaton supported by the extra radial dimension, whose asymptotic value defines
the boundary. Depending on the hierarchy, this boundary can be time-like or
space-like with, in the latter case, potential cosmological applications.
