I am reading Wolff's Note on counterexamples in strong unique continuation problems:
http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf
On Page 3, he wrote "Since $Z_n(\frac {x}{|x|})$" is a solution to a second order ODE, $\nabla Z_n$ vanishes only at 0. I got completely lost in this sentence.
The background is as follows. $Z_n(n\geq 2)$ is the homogeneous term of degree $n$ in the Taylor expansion of the function $\frac {1}{|x-e|^{d-2}}$ (the fundamental solution) at $0$, where $d\geq 3$ is the dimension and $e$ is an arbitrary unit vector in $\mathbb R^d$.
Edited: I deleted a proposition that I thought to be true. It turns out that was false.
