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.