  We discuss constants of motion of a particle under an external field in a
curved spacetime, taking into account the Hamiltonian constraint which arises
from reparametrization invariance of the particle orbit. As the necessary and
sufficient condition for the existence of a constant of motion, we obtain a set
of equations with a hierarchical structure, which is understood as a
generalization of the Killing tensor equation. It is also a generalization of
the conventional argument in that it includes the case when the conservation
condition holds only on the constraint surface in the phase space. In that
case, it is shown that the constant of motion is associated with a conformal
Killing tensor. We apply the hierarchical equations and find constants of
motion in the case of a charged particle in an electro-magnetic field in black
hole spacetimes. We also demonstrate that gravitational and electro-magnetic
fields exist in which a charged particle has a constant of motion associated
with a conformal Killing tensor.
