potential  does not necessarily contain all possible terms. In fact, each term is multiplied by an arbitrary integration constant $\z_i$ that can be set to zero. Such a vanishing $\z_i$ implies, from Eq. , that the corresponding blow-up modulus $z_i$ is constant. Hence, we learn that we can consistently freeze an arbitrary subset of blow-up moduli $z_i$ and thereby select an arbitrary subset of the vectors ${\bf P}_i$ that appear in the potential . Then, for suitable subsets and choices of signs $\m_i$, the associated matrix $A$ can well be proportional to a Cartan matrix, as we will see for several explicit examples studied further below. The system is then Toda and can be integrated analytically. The resulting solutions are, of course, special in that they rely on a number of blow-up moduli being frozen. However, in the present context, this seems to be the best one can do by analytic methods.

Before we go on to analyse explicit examples, it is worth making a general observation about the structure of the solutions. It is well-known [@Lukas:1996ee; @Lukas:1996iq], for Lagrangians of the type , that the fields $z_i$ approach constant values asymptotically at early and late times (both in the 