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The infinite families of Peregrine, Akhmediev and Kuznetsov-Ma breathersolutions of the focusing Nonlinear Schroedinger (NLS) equation are obtainedvia a matrix version of the Darboux transformation, with a spectral matrix ofthe form of a Jordan block. The structure of these solutions is essentiallydetermined by the corresponding solution of the Lyapunov equation. Inparticular, regularity follows from properties of the Lyapunov equation.

NLS breathers, rogue waves, and solutions of the Lyapunov equation for Jordan blocks