  Killing tensor fields have been thought of as describing hidden symmetry of
space(-time) since they are in one-to-one correspondence with polynomial first
integrals of geodesic equations. Many problems in classical mechanics can be
formulated as geodesic problems in curved spaces and spacetimes, and thus
solving the defining equation for Killing tensor fields (the Killing equation)
is a powerful way to integrate the equations of motion. In this paper we
attempt to formulate the integrability conditions of the Killing equation,
which serve to put an upper bound on the number of linearly independent
solutions and also to restrict the possible forms of solutions tightly. To this
end, we first show the prologation for the Killing equation in a manner that
uses Young symmetrizers. Then, using the prolonged equations, we provide the
integrability conditions explicitly.
