Complete set of quasi-conserved quantities for spinning particles around Kerr

We revisit the conserved quantities of Mathisson-Papapetrou-Tulczyjew equations describing motion spinning particles on a fixed background. Assuming Ricci-flatness and existence Killing-Yano tensor, we demonstrate that besides two non-trivial quasi-conserved quantities, i.e. at linear order in spin, found by Rüdiger, are one-to-one correspondence with mixed-symmetry Killing tensors. prove no such stationary axisymmetric tensor exists Kerr geometry. discuss implications for spacetime where quasi-constants shown not to be complete involution.