  Dynamical systems methods are used to investigate domain-wall solutions of a
two-parameter family of models in which gravity is coupled to an axion, and to
a dilaton with an exponential potential of either sign. A complete global
analysis is presented for (i) constant axion and (ii) flat walls, including a
study of bifurcations and a new exact domain-wall solution with non-constant
axion. We reconsider `fake supergravity' issues in light of these results. We
show, by example, how domain walls determine multi-valued superpotentials that
branch at stationary points that are not stationary points of the potential,
and we apply this result to potentials with anti-de Sitter vacua. We also show
by example that `adapted' truncation to a single-scalar model is sometimes
inconsistent, and we propose a `generalized' fake supergravity formalism that
applies in some such cases.
