= Solution
The new <characteristics> have invariant $x_1-x_2$. On the initial curve it is $D(s)=s$ for $s<0$ and $D(s)=s-s^2$ for $s\ge0$. The latter branch folds at $s=1/2$, attaining its maximum $1/4$. Some characteristics meet the initial curve twice or more; generic values of $f$ at those intersections disagree, so no global solution constant along characteristics can exist.
At every initial point except $s=1/2$, the curve is non-characteristic, and the local existence and uniqueness theorem applies. At $s=1/2$, tangency requires $f'(1/2)=0$, but this is not sufficient. The <fold tangency obstruction for characteristic data> requires locally $f(s)=F(D(s))$ with $F$ continuously differentiable: equal invariant values must have equal data, and $f'(s)/(1-2s)$ must have a finite continuous limit. For example $f(s)=|s-1/2|^{3/2}$ satisfies $f'(1/2)=0$ and agrees across the fold, but would force an infinite derivative of $F$ at $1/4$.
If the full compatibility condition holds, $F$ can be extended to invariant values above $1/4$ to give a local solution. Those values are not specified by the initial curve, so local uniqueness then fails.
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