  Recently, the asymptotic behaviour of three-dimensional anti-de Sitter
gravity with a topological mass term was investigated. Boundary conditions were
given that were asymptotically invariant under the two-dimensional conformal
group and that included a fall-off of the metric sufficiently slow to
consistently allow pp-wave type of solutions. Now, pp-waves can have two
different chiralities. Above the chiral point and at the chiral point, however,
only one chirality can be considered, namely the chirality that has the milder
behaviour at infinity. The other chirality blows up faster than AdS and does
not define an asymptotically AdS spacetime. By contrast, both chiralities are
subdominant with respect to the asymptotic behaviour of AdS spacetime below the
chiral point. Nevertheless, the boundary conditions given in the earlier
treatment only included one of the two chiralities (which could be either one)
at a time. We investigate in this paper whether one can generalize these
boundary conditions in order to consider simultaneously both chiralities below
the chiral point. We show that this is not possible if one wants to keep the
two-dimensional conformal group as asymptotic symmetry group. Hence, the
boundary conditions given in the earlier treatment appear to be the best
possible ones compatible with conformal symmetry. In the course of our
investigations, we provide general formulas controlling the asymptotic charges
for all values of the topological mass (not just below the chiral point).
