we will determine later. From Eq.  the equations of motion for the blow-up moduli $z_i$ are given by $$\frac{d}{d\t}\left(e^{-{\bf P}_i\cdot\bph +3\a -\n}\dot{z}_i\right)=0\; ,$$ where the dot denotes the derivative with respect to $\t$. They can be easily integrated to $$\dot{z}_i = \z_i\, e^{{\bf P}_i\cdot\bph -3\a +\n}\; , \label{zsol}$$ with arbitrary integration constants $\z_i$. Let us now choose the particularly convenient gauge $\n = 3\a$. In this gauge, the sum of the $(00)$ component and the spatial components of the Einstein equations reads $\ddot{\a}=0$ and, we have $$\a = v\t +\a_0 \label{a0}$$ where $v$ and $\a_0$ are arbitrary integration constants. Proper time $t$ is obtained by integrating $$dt = e^{3\a}d\t$$ which leads to $$v\t +\a_0 = \frac{1}{3}\ln\left(\sqrt{3E}|t-t_0|\right)\; .
 \label{pt}$$ Here $t_0$ is another integration constant and $$E =3v^2\; .$$ The solution  for the scale factor, written in proper time, then takes the form $$\a = \frac{1}{3}\ln\left(\sqrt{3E}|t-t_0|\right)$$ and shows the expected power-law behaviour, with power $1/3$, characteristic of evolution driven by kinetic energy. As usual, we have two branches, the negative-time branch, $t-t_0<0$, which ends in a future curvature singularity, and the positive-time branch, $t-t_0>0$, which starts out in a past curvature singularity.

We still have to consider the equations of motion 