Good morning,

I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(x) \right)$ for every $x\in \mathbb{R}^n$.

Given a real function $\lambda:\mathbb{R}^n\to \mathbb{R}$, we are interested into the pde $\nabla v(x).v(x)=\lambda(x) v(x)$, in other question we ask that $v$ is an eigenvector of its own Jacobian matrix. Apart from simple cases (e.g. constant functions, or cases where the component $v_i$ depends just on $x_i$) I have not been able to find an exhaustive answer, nor I was able to find references on Google. It is not even clear to me how to treat the case of separable variables up to now.

However, I feel that something general should be known on such equations. Can anybody provide me with some references, or give a hint on how to attack the general problem?

Thank you very much!