The complete analytical solution of the governing nonlinear vector equations is found which describes periodic and solitary propagating-rotating waves in an initially helical fiber. A detailed description of various types of these waves is given. The solitary wave velocity is found to be proportional to the square root of the amplitude of the internal force, and the effective wave length is shown to be independent of the amplitude. The essential influence of the rigid body rotation of the helix on the wave shape is shown. Axial and angular momenta are determined as well.