{\ensuremath{\partial}}_t w= & - \frac{1-\dot{\tau}}{4} e^{\frac{3}{4}s} W + (1-\dot{\tau}) e^{\frac{3s}{4}} {\ensuremath{\partial}}_s W + \dot{\kappa} \label{BUDS:1}
 + \frac{5}{4}(1 - \dot{\tau}) x e^{\frac{3}{4}s} {\ensuremath{\partial}}_x W - \dot{\xi} e^{s} {\ensuremath{\partial}}_x W\,, \\ \label{BUDS:2}
{\ensuremath{\partial}}_\theta w = & e^{ s} {\ensuremath{\partial}}_x W\,. \end{aligned}$$

Next, we record the calculations $$\begin{aligned}
{\ensuremath{\partial}}_t z 
=  (1 - \dot{\tau}) e^s {\ensuremath{\partial}}_s Z + (\frac 5 4 (1 - \dot{\tau}) x e^s - \dot{\xi} e^{\frac 5 4 s}) {\ensuremath{\partial}}_x Z\label{whilk:miskey:1} ,\quad
{\ensuremath{\partial}}_\theta z=   e^{\frac 5 4 s} {\ensuremath{\partial}}_x Z\,. \end{aligned}$$ and similarly, $$\begin{aligned}
{\ensuremath{\partial}}_t a = (1 - \dot{\tau}) e^s {\ensuremath{\partial}}_s A + (\frac 5 4 (1 - \dot{\tau}) x e^s - \dot{\xi} e^{\frac 5 4 s}) {\ensuremath{\partial}}_x A,  \label{whilk:miskey:4}\quad
{\ensuremath{\partial}}_\theta a=  \ e^{\frac 5 4 s} {\ensuremath{\partial}}_x A\,.\end{aligned}$$

Then in self-similar variables becomes $$\begin{aligned}
 {\ensuremath{\nonumber}}&({\ensuremath{\partial}}_s - \frac 1 4) W + \left(\frac 5 4 x - \beta_\tau (\dot{\xi} - \kappa) e^{\frac 1 4 s} +\beta_{\tau}(\beta_2 e^{\frac 1 4 s} Z+ W)\right){\ensuremath{\partial}}_x W  \\ 
&\quad= - \beta_\tau e^{- \frac 3 4 s}\dot{\kappa} -  \beta_\tau e^{- \frac 3 4 s} A \Big( \beta_3 Z + \beta_4 (e^{- \frac s 4}W + \kappa) \Big)\,.\label{eq:W:0}\end{aligned}$$ Similarly, we rewrite as $$\begin{aligned}
&{\ensuremath{\partial}}_s Z + \left(\frac 5 4 x+ \beta_{\tau}(e^{\frac 1 4 s}(\beta_2\kappa-\dot{\xi} +Z)+\beta_2 