A vector-wave analysis of nondiffracting beams propagating along a birefringent chiral crystal for the case of tensor character of both the optical activity and linear birefringence is presented. We have written characteristic equations and found propagation constants and amplitude parameters of the eigenmodes. The characteristic curves have anomalous zones described by an isotropic point or a gap-point, provided that the elements of an optical activity tensor obey the requirement g(11)g(33)<0, |g(33)|>|g(11)|. In the anomalous zone, a nondiffracting beam can propagate through a purely chiral crystal as if through an isotropic medium. We have shown that the field of eigenmodes is nonuniformly polarized in the beam cross section, while the field with the initially uniform polarization distribution experiences periodic transformations. We have revealed that even a purely chiral crystal without linear birefringence can generate optical vortices in an initially vortex-free Bessel beam.