. A non-semisimple matrix loop algebra is presented, and a class of zero curvature
equations over this loop algebra is used to generate bi-integrable couplings. An
illustrative example is made for the Dirac soliton hierarchy. Associated variational identities
yield bi-Hamiltonian structures of the resulting bi-integrable couplings, such that
the hierarchy of bi-integrable couplings possesses infinitely many commuting symmetries
and conserved functionals.