  The phase space of a Hamiltonian system is symplectic. However, the
post-Newtonian Hamiltonian formulation of spinning compact binaries in existing
publications does not have this property, when position, momentum and spin
variables $[X, P, S_1, S_2]$ compose its phase space. This may give a
convenient application of perturbation theory to the derivation of the
post-Newtonian formulation, but also makes classic theories of a symplectic
Hamiltonian system be a serious obstacle in application, especially in
diagnosing integrability and nonintegrability from a dynamical system theory
perspective. To completely understand the dynamical characteristic of the
integrability or nonintegrability for the binary system, we construct a set of
conjugate spin variables and reexpress the spin Hamiltonian part so as to make
the complete Hamiltonian formulation symplectic. As a result, it is directly
shown with the least number of independent isolating integrals that a
conservative Hamiltonian compact binary system with both one spin and the pure
orbital part to any post-Newtonian order is typically integrable and not
chaotic. And conservative binary system consisting of two spins restricted to
the leading order spin-orbit interaction and the pure orbital part at all
post-Newtonian orders is also integrable, independently on the mass ratio. For
all other various spinning cases, the onset of chaos is possible.
