ads cft - Boundary conditions for fields in Kerr/CFT

I am reading a paper by Guica et al. on Kerr/CFT correspondence (arXiv:0809.4266) and I'm not sure if I got this. They choose the boundary conditions, like a deviation of the full metric from the background Near-Horizon Extremal Kerr (NHEK) metric. Let's say that we can write that deviation like

$$\delta_\xi g_{\mu \nu}=\mathcal{L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$$

And the most general diffeomorphism which preserve the boundary conditions given in the text is:

$$\xi=[-r\epsilon'(\varphi)+\mathcal{O}(1)]\partial_r+\left[C+\mathcal{O}\left(\frac{1}{r^3}\right)\right]\partial_\tau+\left[\epsilon(\varphi)+\mathcal{O}\left(\frac{1}{r^2}\right)\right]\partial_\varphi+\mathcal{O}\left(\frac{1}{r}\right)\partial_\theta$$

What my mentor told me, while briefly explaining this, is that we basically need to find the most general $\xi$ such that $\mathcal{L}_\xi g_{\mu\nu}$ is within the class of the boundary conditions.

But how do I find these $\xi$?

How can you find these boundary conditions and diffeomorphisms? Or better jet, how do I find diffeomorphism using those boundary conditions? :\