The nature of a singularity in a cool, gravitation-free axisymmetrical, relativistic, steady pulsar wind is further investigated, in line with the recent counter-argument of Ardavan (1995, hereafter A95) that the pure Alfvén ‘singularity’ has an equal importance to the Alfvén singularity if one defines the singularity via a quadratic form rather than using the ratio as adopted by Li & Melrose (1994). The pure Alfvénic point in A95 coincides with the proposed characteristic point through which a continuous wind will meet the star and infinity. However, we find that the critical point as implied in the quadratic form is in fact the intermediate point (Ardavan 1979), but not the pure Alfvénic point. Thus the analysis of A95 does not indicate any significance of the pure Alfvénic point. We also demonstrate that the intermediate point which appears in the quadratic form of A95 is not genuine.