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We prove the following result: if a continuous vector field $F$ is Lipschitzwhen restricted to the hypersurfaces determined by a suitable foliation and atransversal condition is satisfied at the initial condition, then $F$determines a locally unique integral curve. We also present some illustrativeexamples and sufficient conditions in order to apply our main result.

A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs