System: $$\frac{\partial u}{\partial x}=v,\frac{\partial u}{\partial t}=w$$$$\frac{\partial v}{\partial x}=G(x,t),\frac{\partial v}{\partial t}=-\dot{a}(t)G(x,t)$$$$\frac{\partial w}{\partial x}=-\dot{a}(t)G(x,t),\frac{\partial w}{\partial t}=G(x,t)$$
For $G(x,t)=g''(x-a(t))$, with $g$ and $a$ smooth.
Question: Using Frobenius theorem, show the system has a solution if $\dot{a}^2(t)=1$.
Comments: Generally for this types of question I use $[\mathbb{X},\mathbb{Y}]=0$ as the condition for the system to have a solution.
Is it possible to use that relationship here?
Haven't been able to make much progress don't know where to start, so any help on how to do questions of this form would be greatly appreciated.