Exterior Schwarzschild metric
$$ ds^2=\Big(1-\frac{2M}{r}\Big)dt^2- \Big(1-\frac{2M}{r}\Big)^{-1}dr^2-r^2 d\theta^2-r^2\sin^2\theta d\phi^2 $$
I knew that there are two Killing vectors associated with the Schwarzschild metric, $K^{(1)}=(1, 0, 0, 0)$ and $K^{(2)}=(0, 0, 0, 1)$.
But, in an article written that there are four Killing vectors in Schwarzschild metric. $$ K^{(1)}_\mu=\Big(1-\frac {2M}{r}\Big)\delta^t_\mu $$
$$ K^{(2)}_\mu=r^2\sin^2\theta \delta^\phi_\mu $$
$$ K^{(3)}_\mu=r^2(\sin \phi\delta^\theta_\mu+\sin\theta\cos\theta\cos\phi\delta^\phi_\mu) $$
$$ K^{(4)}_\mu=r^2(\cos \phi\delta^\theta_\mu-\sin\theta\cos\theta\sin\phi\delta^\phi_\mu) $$
corresponding to time translations and infinitesimal spatial rotations.
I did not understand how are Killing vectors defined? Why are they found differently from classical GR books? What is the method to obtain the Killing vectors of arbitrary metric?
If the metric does not depend on $x^k$ coordinate $K^\mu=\delta^\mu_k$ is a Killing vector. The covariant components which are $ K_\mu=g_{\mu\nu}K^\nu=g_{\mu\nu}\delta^\nu_k $. $$ K^{(1)}_\mu=g_{\mu\nu}\delta^\nu_k=g_{k\nu}\delta^\nu_\mu=g_{tt}\delta^t_\mu $$
$$ K^{(2)}_\mu=g_{\mu\nu}\delta^\nu_k=g_{k\nu}\delta^\nu_\mu=g_{\phi\phi}\delta^\phi_\mu $$
This is understandable. But how are obtained the second and third Killing vectors? If they are rotation associated vectors around which axis rotation carried out?
I found the solutions for $\xi_\theta$, $\xi_\phi$ from the Killing equations for the metric of the $2D$ sphere.
But, how they are picking the arbitrary values for $A$, $B$, $C$ is unclear.