= Solution
The physically meaningful extension requires $h\geq0$: negative net input removes water where wet and cannot keep removing nonexistent surface water where dry. Keeping the algebraic source $E(t)<0$ unchanged in a dry zone would instead produce negative depths, so the drying-front description implicitly includes this availability condition.
Use the same $H(t)$, increasing until $t^*$ and decreasing afterwards. While $H(t)>0$, there is one earlier wetting time $\tau_d(t)\in[0,t^*]$ such that $H(\tau_d(t))=H(t)$. The boundary-emitted <characteristic> with this label has just exhausted its water. Its position is the <drying front>:
$$
\boxed{s_d(t)=\frac53\int_{\tau_d(t)}^t[H(r)-H(t)]^{2/3}\,dr.}
$$
The integrand is nonnegative because $H(r)\geq H(t)$ between these two equal-height times. At $t=t^*$ the interval collapses and $s_d=0$; as net input becomes negative the front moves downstream. Differentiating gives $s_d'=-(10/9)E(t)\int_{\tau_d(t)}^t[H(r)-H(t)]^{-1/3}\,dr>0$, where endpoint singularities are integrable away from the turning-time limit.
The wet solution ahead of the front still follows unexhausted <characteristics>. With $S_0(t)=\frac53\int_0^tH(r)^{2/3}\,dr$,
$$
\boxed{h(s,t)=\begin{cases}0,&0\leq s\leq s_d(t),\\H(t)-H(\tau(s,t)),&s_d(t)<s<S_0(t),\\H(t),&s\geq S_0(t),\end{cases}}
$$
where $0\leq\tau<\tau_d(t)$ is determined by the same <characteristic>-position <integral> as in (iv). If cumulative loss reaches $H(t)=0$, all remaining initial-line water has also vanished and the surface is dry thereafter while $E<0$. If the total later loss is smaller, the formula applies for all later times with $H(t)>0$. This is the <drying front in a rainfall-driven Manning kinematic wave>.
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