  A Bargmann symmetry constraint is proposed for the Lax pairs and the adjoint
Lax pairs of the Dirac systems. It is shown that the spatial part of the
nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Liouville
integrable Hamiltonian system and that under the control of the spatial part,
the time parts of the nonlinearized Lax pairs and adjoint Lax pairs are
interpreted as a hierarchy of commutative, finite dimensional Liouville
integrable Hamiltonian systems whose Hamiltonian functions consist of a series
of integrals of motion for the spatial part. Moreover an involutive
representation of solutions of the Dirac systems exhibits their integrability
by quadratures. This kind of symmetry constraint procedure involving the
spectral problem and the adjoint spectral problem is referred to as a binary
nonlinearization technique like a binary Darboux transformation.
