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It is shown that there exists an exact paraxial cold-fluid equilibrium of a high-intensity, space-charge-dominated charged-particle beam with a periodically twisted elliptic cross section in a nonaxisymmetric periodic magnetic field. Generalized envelope equations, which determine the beam envelopes, ellipse orientation, density, and internal flow velocity profiles, are derived. Nonrelativistic and relativistic examples of such beam equilibria are presented. The equilibrium and stability of such beams are demonstrated by self-consistent particle-in-cell (PIC) simulations.