  In general relativity, the motion of an extended body moving in a given
spacetime can be described by a particle on a (generally non-geodesic)
worldline. In first approximation, this worldline is a geodesic of the
underlying spacetime, and the resulting dynamics admit a covariant and
4-dimensional Hamiltonian formulation. In the case of a Kerr background
spacetime, the Hamiltonian was shown to be integrable by B.~Carter and the now
eponymous constant. At the next level of approximation, the particle possesses
proper rotation (hereafter \textit{spin}), which couples the curvature of
spacetime and drives the representative worldline away from geodesics. In this
article, we lay the theoretical foundations of a series of works aiming at
exploiting the Hamiltonian nature of the equations governing the motion of a
spinning particle, at linear order in spin. Our formalism is covariant and
10-dimensional. It handles the degeneracies inherent to the local Lorentz
invariance of general relativity with tools from Poisson geometry, and accounts
for the center-of-mass/spin-supplementary-condition using constrained
Hamiltonian system theory. As a first application, we consider the
linear-in-spin motion in a Kerr background. We show that the resulting
Hamiltonian system admits exactly five functionally independent integrals of
motion related to Killing symmetries, thereby proving that the system is
integrable. We conclude that linear-in-spin corrections to the geodesic motion
do not break integrability, and that the resulting trajectories are not
chaotic. We explain how this integrability feature can be used to reduce the
computational cost of waveform generation schemes for asymmetric binary systems
of compact objects.
