It is known that coordinate transformation generated by Killing vectors (KV) preserve the metric components, i.e. it generates an isometry transformation. Are there similar geometrical quantities that are preserved by Killing tensors?
Symmetrized product of Killing vectors can behave as Killing tensors, however, not all Killing tensors can be decomposed in such ways. For e.g. take $K_{ab}=\xi^1_{(a}\xi^2_{b)}+g_{ab}$ which is still a Killing 2-tensor and for $\xi^1_a=\xi^2_a$, $K_{ab}$ behaves as induced metric on surface orthogonal to $\xi_a$. Even if we know what an individual KV preserves, it is still not intuitively clear what quantities are preserved by products of KVs, let alone a general Killing tensor.
A general space-time need not admit Killing tensors, just like KVs. So we can try to look at asymptotic limit (say asymptotically flat space-time) and ask if there are "BMS tensors" as an extension for BMS vectors. However to derive such asymptotic tensors we will need to look at the invariant quantities preserved by Killing tensors at infinity and then transform it in the unphysical coordinates on $\mathscr{I}$ (as it was done for BMS vectors here). My guess was that a Killing tensor of type $(0,p)$ should preserve a $p-$dimensional manifold, however, I am not sure if this correct or how to approach this question.
