geometric interpretation of eigenfunctions of a vector field

Question

Let $M$ be a smooth manifold and $X\in\mathfrak X(M)$ be a section on the tangent bundle.

What is the geometrical interpretation of the eigenfunctions of $X$. That is functions in $f\in \mathcal C^\infty$ with $X[f]=\lambda f$, where $\lambda$ is in the reals?

What is the geometrical interpretation of the spectrum of $X$?