Nonlinear wave vector solutions of equations describing the motion of a magnetized hollow elec- tron beam in a vacuum drift tube are derived. In the case of an arbitrary radical distribution of the beam density in the initial equilibrium state, the wave solutions written in an integral form are multipole helical vortices fill- ing the entire cross section of the beam. For the hollow beam with uniform density in the equilibrium state, the wave solution is obtained in an algebraic form as axially uniform multipole vortices similar to axially uniform vortices in a rotating magnetized neutral or purely electron bounded plasma.
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