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This note serves to provide additional details for the proof of Lemma 3.6 inour paper [Liu, Zhang and Zhang, Comm. Math. Sci., 3(2005), pp.201-218].Moreover, we will also present an alternative, yet simpler, proof based onarguments in [Wang, Zhang and Zhang, CPAM, 68(2015), no. 8, 1326-1398].

A note to paper: Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential