Geometric Interpretation of the Jacobian Matrix and Its Eigenvectors

I understand that for scalar-valued functions $g: \mathbb{R}^n \to \mathbb{R}$, the gradient represents the direction of maximum ascent. Similarly, for vector-valued functions $f: \mathbb{R}^n \to \mathbb{R}^n$, each row of the Jacobian matrix $J_f(x)$ provides the direction of maximum ascent for the corresponding component function.

I have the following questions:

1. Geometric interpretation of eigenvectors of the Jacobian matrix: What is the geometric interpretation of the eigenvectors of the Jacobian matrix at a point? How do they affect the behavior of the function near that point?

2. Same eigenvectors at every point: Suppose the Jacobian matrix of $f$ has the same set of eigenvectors at every point in $\mathbb{R}^n$. Does this imply that the function is linear or Jacobian diagonal, or if not is there some special structure to the function $f$? What are the nontrivial examples of vector fields with same eigen vectors at every point in $\mathbb{R}^n$

Any insights or references would be greatly appreciated!