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 $X = a_i x^i \partial_i + O(|x|^2)$ 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 functions $f \in C^{\infty}(U,\mathbb{R})$ and all $\alpha \in \mathbb{R}$ such that
$$ \partial_X f = \alpha \cdot f$$
on some neighborhood $U$ of $0$.

**Conjecture:** From looking at the linear example $X = a_ix^\partial_i$, I would think that either $f = 0$ or $\alpha = \sum k_i a_i$ for some $k \in \mathbb{N}_0^n$ and $f = C\cdot x^k + O(|x|^{k+1})$. (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.

2) 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.

/Edit: Renamed $\eta$ to $f$.