  We establish the existence of a deformation of the usual Carter constant
which is conserved along the motion in a fixed Kerr background of a spinning
test body possessing the spin-induced quadrupole coupling of a black hole. The
conservation holds perturbatively up to second order in the test body's spin.
This constant of motion is obtained through the explicit resolution of the
conservation constraint equations, employing covariant algebraic and
differential relations amongst covariant building blocks of the Kerr
background. For generic spin-induced quadrupole couplings, which describe
compact objects such as neutron stars, we obtain a no-go result on the
existence of such a conserved quantity.
