  We show that the Killing tensor of the Kerr spacetime has an analogue in the
$\sqrt{\rm Kerr}$ gauge theory solution related to it by the classical double
copy. This hidden symmetry of $\sqrt{\rm Kerr}$ leads to an additional constant
of motion for color-charged point particles moving in it, implying
integrability of the equation of motion. These are the gauge theory
counterparts to the Carter constant and the integrability of the geodesic
equation in a Kerr background.
