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A non-classical Weyl theory is developed for skew-self-adjoint Dirac systemswith rectangular matrix potentials. The notion of the Weyl function isintroduced and direct and inverse problems are solved. A Borg-Marchenko typeuniqueness result and the evolution of the Weyl function for the correspondingfocusing nonlinear Schr\"odinger equation are also derived.

Skew-self-adjoint Dirac systems with a rectangular matrix potential: Weyl theory, direct and inverse problems