of motion . The reason behind this is that not all bulk moduli are on the same footing because of their differing couplings to the blow-up moduli, as is evident from the coupling vectors in Table 1. Instead, we start with the slightly more general Ansatz $$\f_A = c_A\f \label{A1}$$ where $\f$ is the breathing mode and $c_A$ are constants to be determined. For this Ansatz to be successful we need the seven equations of motion  for $\f_A$ to reduce to a single equation for $\f$. This leads to the conditions $$\begin{aligned}
 {\bf p}_{(\t ,a)}\cdot{\bf c} &=& {\rm const} \label{c1} \\
 \sum_{\t ,a}\z_{(\t ,a)}\, p_{(\t ,a)A} &=& \frac{1}{2}c_A\z^2 \label{c2}\end{aligned}$$ where we have defined $$\z_{(\t ,a)}^2 = \sum_n \z_{(\t ,n,a)}^2\; ,$$ and $\z$ is a constant. These conditions lead to a unique solution $${\bf c} = \frac{2}{7}(4,4,4,3,3,3,3)$$ for the vector ${\bf c}$ in our Ansatz . It can further be shown that all conditions , can be satisfied for specific choices of coefficients $\z_{(\t ,a)}$ within a three-parameter family of solutions. The details of this are inessential for our subsequent discussion. For such a solution, the seven equations of motion  reduce to the single equation $$\frac{1}{2}\ddot{\f}-2\m\z^2 e^{-2\m\f} = 0\; ,$$ where 