Consider a Riemannian manifold $(M^n , g)$ and let $d_p: M^n \to [0,\infty)$ be the distance function of $p \in M^n$. Then the flow lines generated by $\nabla d_p$ are radial geodesics from $p$. Also, each geodesic sphere $S_r(p)$ is orthogonal to the flow lines.

I wonder for a general vector field $X$, can one find a "natural" section of the flow lines? For example, if we assume $X$ is a Killing vector field, can one construct $\Sigma_t$ that is orthogonal to every flow line and the flow maps each $\Sigma_t$ to another?