  A point-like object moving in a background black hole spacetime experiences a
gravitational self-force which can be expressed as a local function of the
object's instantaneous position and velocity, to linear order in the mass
ratio. We consider the worldline dynamics defined by the conservative part of
the local self-force, turning off the dissipative part, and we ask: Is that
dynamical system a Hamiltonian system, and if so, is it integrable?
  In the Schwarzschild spacetime, we show that the system is Hamiltonian and
integrable, to linear order in the mass ratio, for generic (but not necessarily
all) stable bound orbits. There exist an energy and an angular momentum, being
perturbed versions of their counterparts for geodesic motion, which are
conserved under the forced motion. We also discuss difficulties associated with
establishing analogous results in the Kerr spacetime. This result may be useful
for future computational schemes, based on a local Hamiltonian description, for
calculating the conservative self-force and its observable effects. It is also
relevant to the assumption of the existence of a Hamiltonian for the
conservative dynamics for generic orbits in the effective-one-body formalism,
to linear order in the mass ratio, but to all orders in the post-Newtonian
expansion.
