an $SU(2)^3$ Toda model with three exponentials appearing in the potential . From Table 1 there are a number of other options and in the following we will simply assume the existence of $n$ vectors ${\bf P}_i$ satisfying  to cover all possibilities.

The standard procedure to deal with such a system is to introduce a new constant basis ${\bf e}_A=({\bf e}_i,{\bf e}_a)$ in field space, where $i =1,\cdots ,n$ and $a=n+1,\cdots ,7$ such that $${\bf e}_i = \frac{1}{\sqrt{2}}{\bf P}_i$$ and the remaining vectors ${\bf e}_a$ are chosen to complete the set to an orthonormal basis satisfying $${\bf e}_A\cdot{\bf e}_B = \d_{AB}\; .$$ The bulk fields $\bph$ can then be expanded as $$\bph = \sum_A\r_A{\bf e}_A\; ,$$ where $\r_A$ is a new set of fields. The Lagrangian  and the Hamiltonian , written in terms of these new fields, take the form $$\cL = \frac{1}{4}\sum_A\dot{\r}_A^2-V\; ,\qquad
 \cH = \frac{1}{4}\sum_A\dot{\r}_A^2+V=E\; , \label{LH}$$ with the potential $$V = 4\sum_i\z_i^2e^{2\r_i}\; .$$ Note that the modes $\r_a$ have decoupled from this potential, which is, of course, one of the motivations behind introducing the basis ${\bf e}_A$.

The general solution to the Lagrangian in  can be easily obtained as $$\bph = \sum_A\r_A{\bf e}_A$$ with $$\begin{aligned}
 \r_a &=& 