  A simple proof is given of the classical result due to Fatkullin and
Slastikov (2005), Liu, Zhang and Zhang (2005) that critical points for the
Onsager functional with the Maier-Saupe molecular interaction are axisymmetric,
including the case of stable critical points with an additional dipole-dipole
interaction (Zhou et al 2007). The proof avoids spherical polar coordinates,
instead using an integral identity on the sphere $S^2$. For general
interactions with absolutely continuous kernels the smoothness of all critical
points is established, generalizing a result of Vollmer (2016) for the Onsager
interaction. It is also shown that non-axisymmetric critical points exist for a
wide variety of interactions including that of Onsager.
