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You get a separate ODE for every integral curve (which you denote by $\gamma$) of the vector field $X$. The space $\mathbb R^4$ is a disjoint union of these integral curves. Each ODE is of order one, so you need to fix the value of $f$ at exactly one point from every integral curve. (Alternatively, you could also take asymptotic behaviour on each integral curve as a boundary condition.) You can fix the values on a three dimensional surface that meets every integral curve exactly once, provided such a surface exists. A good choice for such a surface or a collection of such surfaces depends on the vector field $X$.