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We show that hydrodynamic theories of polar active matter generically possessinhomogeneous traveling solutions. We introduce a unifying dynamical-systemframework to establish the shape of these intrinsically nonlinear patterns, andshow that they correspond to those hitherto observed in experiments andnumerical simulations: periodic density waves, and solitonic bands, orpolar-liquid droplets both cruising in isotropic phases. We elucidate theirrespective multiplicity and mutual relations, as well as their existencedomain.

Emergent spatial structures in flocking models: a dynamical system insight