Killing tensor and Conserved quantities 
The definition of the killing tensor is written above, as taken from Wikipedia. My question here is two-fold:


*

*Can all Killing tensors be build from the Killing vectors of that spacetime?

*Do Killing tensors also correspond to conserved quantities? If yes, are they independent of whatever are the conserved quantities found from Killing vectors of that spacetime?
 A: 
Can all Killing tensors be build from the Killing vectors of that spacetime?

Not always. A classic result by Walker and Penrose:


*

*Walker, M., & Penrose, R. (1970). On quadratic first integrals of the geodesic equations for type {22} spacetimes. Communications in Mathematical Physics, 18(4), 265-274, doi:10.1007/BF01649445,


demonstrated, in particular, that Kerr spacetime has a rank 2 Killing tensor independent of the Killing vectors of this spacetime.

Do Killing tensors also correspond to conserved quantities?

Yes, one can contract Killing tensor $K^{i\ldots j}$ with the momenta of the geodesic motion $p_i$ forming a scalar,
$$
 I_K = K^{i\ldots  j}p_i \cdots p_j\,.
$$
This scalar would be a conserved quantity of the geodesic motion, so its Poisson bracket with the Hamiltonian $H=g^{ij}p_ip_j$ of the geodesic motion vanishes:
$$ 
\{H,I_K\}=0.
$$

If yes, are they independent of whatever are the conserved quantities found from Killing vectors of that spacetime?

If Killing vectors do not provide enough conserved quantities, then yes, Killing tensors could be sources of independent integrals of motion. See, for example the paper:


*

*Krtouš, P., Kubiznák, D., Page, D. N., & Frolov, V. P. (2007). Killing-Yano tensors, rank-2 Killing tensors, and conserved quantities in higher dimensions. Journal of High Energy Physics, 2007(02), 004, doi:10.1088/1126-6708/2007/02/004, arXiv:hep-th/0612029,


for the discussion of constants of geodesic motion from Killing tensors, with a motion in $D$-dimensional Kerr–NUT–AdS as an example.
