= Solution
The <characteristics> solve $\dot x_j=a_j(x)$ and $\dot u=b(x,u)$, starting with $x$ on the initial hypersurface and $u=\varphi(x)$. The <non-characteristic hypersurface> condition at a point is $a\cdot n\ne0$, where $n$ is any nonzero normal there. It makes the map from initial-surface coordinates and characteristic time to $x$ locally invertible. Local uniqueness for the <ordinary differential equations> and the <inverse function theorem> then give a unique local $C^1$ solution near that point, for the stated $C^1$ coefficients and data.
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