  The motion of a stellar compact object around a supermassive black hole can
be approximated by the motion of a spinning test particle. The equations of
motion describing such systems are in general non-integrable, and therefore,
chaotic motion should be expected. This article discusses the integrability
issue of the spinning particle for the cases of Schwarzschild and Kerr
spacetime, and then it focuses on a canonical Hamiltonian formalism where the
spin of the particle is included only up to the linear order.
