of analysing the evolution of $43$ freely rolling $G_2$ moduli to one that only involves the seven bulk moduli subject to the potential . The remaining problem is, therefore, to solve the seven equations of motion  for the bulk moduli and the constraint . Each solution for the bulk moduli can then be inserted into Eq. , which determines the corresponding evolution of the blow-up moduli.

There exists a well-developed solution theory [@Kostant:qu], [@Lu:1996hh]–[@Lukas:1996zq] for the Lagrangian  and the associated equations of motion  for the bulk moduli if the system is of Toda type. To discuss this more specifically we introduce the matrix $$A_{ij}={\bf P}_i\cdot{\bf P}_j\; , \label{A}$$ which consists of one or more irreducible blocks along the diagonal. The system is called Toda if each of these blocks is proportional to the Cartan matrix of a simple Lie-group. One can now inspect the vectors ${\bf P}_i=-\m_i{\bf p}_i$ from Table 1, where we recall that the $\m_i$ are arbitrary signs. It is clear that the complete nine-dimensional matrix obtained from  is not proportional to a Cartan matrix for any choices of the signs $\m_i$ and, hence, the system is not of Toda type in this case. However, note that the 