  We update and detail the formulation of the duality-invariant systems of N
interacting abelian gauge fields with N auxiliary bispinor fields added. In
this setting, the self-duality amounts to U(N) invariance of the nonlinear
interaction of the auxiliary fields. The U(N) self-dual Lagrangians arise after
solving the nonlinear equations of motion for the auxiliary fields. We also
elaborate on a new extended version of the bispinor field formulation involving
some additional scalar auxiliary fields and study U(N) invariant interactions
with derivatives of the auxiliary bispinor fields. Such interactions generate
higher-derivative U(N) self-dual theories.
