We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of $-frac{1}{2}$-Killing spinors on the round sphere $mathbb{S}^3$.
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