  The aim of this short note is to announce the existence of a one-parameter
family of left-invariant metrics on $S^3$ admitting WK-spinors. This family
contains the two non-Einstein Sasakian metrics with WK-spinors on $S^3$, but
does not contain the standard sphere $S^3$ with Killing spinors. Moreover, any
simply-connected, complete Riemannian manifold $X^3 \not= S^3$ with WK-spinors
such that the eigenvalues of the Ricci tensor are constant is isometric to a
space of this one-parameter family.
