= Solution
Use $(y,w)$ to label the initial point in <phase space>. The <characteristic equations for a transport equation> are $\dot X=a(V)$ and $\dot V=0$, with $X(s)=y$ and $V(s)=w$. Since the <velocity> remains fixed, the <ordinary differential equations> integrate directly:
$$
\boxed{S_{s,t}(y,w)=(y+(t-s)a(w),w).}
$$
This <characteristic flow map> exists for all real times, even if $a$ grows rapidly at infinity: each trajectory samples just one fixed velocity. Its inverse is $S_{t,s}$, and its <Jacobian matrix> is the block triangular matrix
$$
D S_{s,t}=
\begin{pmatrix}I&(t-s)Da(w)\\0&I\end{pmatrix}.
$$
Consequently its <Jacobian determinant> is one. The nonzero <Jacobian determinant> of $a$ is unnecessary for this construction.
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