Stringy Toda cosmologies

We discuss a particular stringy modular cosmology with two axion fields in seven space-time dimensions, decomposable as a time and two flat three-spaces. The effective equations of motion for the problem are those of the SU(3) Toda molecule and, hence, are integrable. We write down the solutions, and show that all of them are singular. They can be thought of as a generalization of the pre-big-bang cosmology with excited internal degrees of freedom, and still suffering from the graceful exit problem. Some of the solutions, however, show a rather unexpected property: some of their spatial sections shrink to a point in spite of winding modes wrapped around them. We also comment how more general, anisotropic solutions, with fewer Killing symmetries, can be obtained with the help of $\mathrm{STU}$ dualities.