This is the first part of a series of work aiming at tackling the issue of
    integrability, or lack thereof, for spinning particles around black hole
    spacetimes in general relativity. In this article, we lay the foundations of
    this program, and present a Hamiltonian system that describes the evolution
    equations for a dipolar particle moving in a background spacetime in general
    relativity. First, we construct a set of symplectic variables that covers the
    phase space of such system. The formulae are valid irrespective of the choice
    of background spacetime, of orthonormal tetrad that defines the spin variables,
    and, most importantly, do not rely on any spin supplementary conditions (or
    whether one has been imposed or not). Second, this formalism is simplified with
    the help of the Tulczyjew-Dixon spin supplementary condition, which is argued
    to be the most convenient one in the present context. This allows us to present
    a 5-dimensional, fully covariant Hamiltonian that generates the linear-in-spin
    dynamics. Third, as an application of our general results, we prove the
    integrability, in the sense of Liouville, of the motion of a particle around a
    Schwarzschild black hole, at linear order in spin. Our formalism opens the door
    to a number of applications and extensions that are surveyed in a dedicated

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