Dispersion properties of eigenwaves of an anisotropic cylindrical solid-state waveguide without frequency dispersion in permittivity tensor components have been analyzed theoretically. The classification of waves has been performed. E type bulk-surface eigenwaves with negative group velocities have been found. The conditions for their existence have been determined. The existence in the waveguide of E-type surface eigenwaves and pseudo-surface eigenwaves of E- and H-types has been shown.