= Solution
The flow of the <vector field> is obtained by solving
$$
\frac{d\widetilde x}{d\varepsilon}=\xi(\widetilde x,\widetilde u),\qquad\frac{d\widetilde u}{d\varepsilon}=\eta(\widetilde x,\widetilde u),\qquad(\widetilde x,\widetilde u)|_{\varepsilon=0}=(x,u).
$$
Define $g^\varepsilon(x,u)$ to be this solution at parameter $\varepsilon$. Uniqueness implies $g^0=\operatorname{id}$ and $g^{\varepsilon+\delta}=g^\varepsilon\circ g^\delta$ wherever both sides are defined: shifting the solution parameter gives the same initial-value problem. In particular $g^{-\varepsilon}$ is the local inverse. This constructs the <one-parameter group> generated by the <vector field>, generally as a local group if the flow is not complete.
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