I am pretty sure this should be text book material, but I couldn't find this anywhere; maybe I just don't know where to look.
Problem: Suppose we have a smooth vector field $X = a_i x^i \partial_i$ with $a_i>0$, defined on some neighborhood of $0$ (or near some point on a manifold, that doesn't really matter since this is a local problem). Now I want to find all smooth function $\eta$ and all $\alpha \in \mathbb{R}$ such that $$ \partial_X \eta = \alpha \cdot \eta$$ on some neighborhood of $0$.
Conjecture: From looking at the linear example $X = a_ix^\partial_i$, I would think that either $\eta = 0$ or $\alpha = \sum k_i a_i$ for some $k \in \mathbb{N}_0^n$ and $\eta = C\cdot x^k + O(|x|^{k+1})$. (Because the exponent has to be integral for the solution to be smooth at $0$).
Remark: 1) This seems to be an eigenvalue problem of some sort, but I have no idea how to make this rigourous; like, in what space does the operator work, what is the domain etc.
- This is of course an ODE along the integral curves of the vector field $X$. The problem is though, that the curves don't start at the point $0$, just start there "asymptotically". One could transfer this into a singular ODE (with which substitution?) and try to go with integrating factors, but this gets tricky.