New Solutions of the Einstein-Dirac Equation in Dimension n=3

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.