  The stability of the kinks of the non-linear ${\mathbb S}^2$-sigma model
discovered in Phys. Rev. Lett. 101(2008)131602 is discussed from several points
of view. After a direct estimation of the spectra of the second-order
fluctuation operators around topological kinks, first-order field equations are
proposed to distinguish between BPS and non-BPS kinks. The one-loop mass shifts
caused by quantum fluctuations around the topological kinks are computed using
the Cahill-Comtet-Glauber formula proposed in Phys. Lett. 64B(1976)283. The
(lack of) stability of the non-topological kinks is unveiled by application of
the Morse index theorem. These kinks are identified as non-BPS states and the
interplay between instability and supersymmetry is explored.
