$y_i\ll -1$ or $y_i\gg 1$. For a valid solution, all $\e_i$ need to be small, which amounts to arranging an overlap between those regions by choosing the time shifts $\t_i$ appropriately. Outside this overlap region control of the approximation is lost as one or more blow-ups become large and take up a significant part of the internal space.

There are also cases leading to a Toda model associated with a higher-rank simple group. Consider, for example, the three vectors $$\begin{aligned}
 {\bf P}_{(\a ,1)} &=& (-1,0,0,0,0,-1,0) \label{P1}\\
 {\bf P}_{(\b ,1)} &=& (1,0,0,0,0,0,1) \label{P2}\\
 {\bf P}_{(\g ,3)} &=& (0,-1,0,0,0,0,-1)\; ,\label{P3}\end{aligned}$$ obtained from Table 1 with the sign choice $\m_{(\a ,1)}=-1$, $\m_{(\b ,2)}=1$ and $\m_{(\g ,3)}=-1$ and set all blow-up moduli with $(\t ,a)$ different from the above to constants. The matrix $A$ in Eq.  is then given by $$(A_{\t\s}) = \left(\begin{array}{rrr}2&-1&0\\-1&2&-1\\0&-1&2\end{array}\right)\; ,$$ which is precisely the Cartan matrix of $SU(4)$. Hence, we are dealing with an $SU(4)$ Toda model.

As before, the first step in solving this model is to introduce a new basis $({\bf e}_A)=({\bf e}_\t ,{\bf e}_a)$, where $\t = 1,2,3$ and $a=4,\dots ,7$, in field space. Here the three vectors ${\bf e}_\t$ are identified with – and the remaining 