No hair theorem and Killing tensors I have 2 questions regarding Killing Tensors : 


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*A practical question is how to guess whether a spacetime has Killing tensors or not. We can guess some simple Killing vectors by looking at the isometry of the metric components. Is there any such simple intuition behind finding Killing tensors for a given metric?

*The no hair theorem states that for static spacetimes we would only have three conserved quantities - Charge, Mass and Angular momentum. What about the conserved quantities from the Killing tensors? Does the no-hair theorem take Killing tensors into account?
 A: 
A practical question is how to guess whether a spacetime has Killing tensors or not. 

This is very hard and in general case open problem, however for some situations (such as vacuum stationary and axially symmetric solutions) there are techniques. See e.g. this paper for the description of an algorithm (Cartan–K̈ähler or prolongation-projection method) with several examples. Since the calculations are quite involved, a practical first step would be to check for some numerical criteria of integrability for the geodesics in such a spacetime (such as plotting Poincaré sections).

The no hair theorem states that for static spacetimes we would only have three conserved quantities - Charge, Mass and Angular momentum. What about the conserved quantities from the Killing tensors? 

No-hair theorems like this one, restricts the number of parameters characterizing a spacetime (vacuum or non-vacuum with a fixed matter content)  with given properties (stationary, with an event horizon). Killing tensors allow us to write conserved quantities for geodesics in a given spacetime, so they offer a parametrization of completely different problem. 
