  We elaborate on integrable dynamical systems from scalar-gravity Lagrangians
that include the leading dilaton tadpole potentials of broken supersymmetry. In
the static Dudas-Mourad compactifications from ten to nine dimensions, which
rest on these leading potentials, the string coupling and the space-time
curvature become unbounded in some regions of the internal space. On the other
hand, the string coupling remains bounded in several corresponding solutions of
these integrable models. One can thus identify corrected potential shapes that
could grant these features generically when supersymmetry is absent or
non-linearly realized. On the other hand, large scalar curvatures remain
present in all our examples. However, as in other contexts, the combined
effects of the higher-derivative corrections of String Theory could tame them.
