= Solution
The reduced mass equation gives $hu=1$ after normalizing the reference discharge. The reduced momentum equation integrates to $\eta+F^2u^2/2=1+F^2/2$ with the reference free-surface level one. Since $h=1/u$,
$$
\boxed{s=1-\frac1u+\frac{F^2}{2}(1-u^2).}
$$
Also $hu=1$ reduces the <concentration> equation to
$$
\boxed{c_x=E^*(u)-c=-s_t.}
$$
With $E^*(1)=1$, a steady uniform reference state is $\boxed{u=h=c=\eta=1,\qquad s=0}$. The integration constants above specify this reference normalization; a different imposed discharge or head changes them.
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