I am trying to understand which are the conformal Killing Fields on the Schwarzschild spacetime. I say that $X$ is a conformal Killing field on $S$ ($S$ is Schwarzschild) if there exists a function $f: S \to \mathbb{R}$ such that \begin{equation} \mathcal{L}_X g = fg, \end{equation} where $g$ is the Schwarzschild metric, and $\mathcal{L}$ is the Lie derivative.
I know that the time translation, $\partial_t$, and the rotations, $\Omega_{ij}$ are Killing fields, therefore conformal Killing fields, with $f$ constant and equal to $0$.
In the Minkowski spacetime, for example, parametrized by $(x^0, x^1, x^2, x^3)$, I know that the ``dilation'' field, i.e. the field $$ \sum_{i=0}^3 x^\lambda \partial_{x^\lambda} $$ is a conformal Killing field, with $f=2$. I would like to understand if there is an analogous in Schwarzschild.