I am have not been following the subject recently, but I think, the existence of a Ricci flat $n$-sphere is an open problem for $>3$. I think any such metric will necessarily have generic holonomy, and no examples of compact simply-connected Ricci-flat manifolds generic holonomy are known. See references in mathoverflow.net/questions/16818/… and especially Berger's "Panaramic view of Riemannian geometry".
– Igor BelegradekJun 18 '11 at 3:11

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It seems that such metric does not admit an isometric $S^1$-action. I.e. the metric if exists has almost no symmetry. Therefore it is unlikely that one can construct it.
– Anton PetruninJun 18 '11 at 16:07

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@Igor, you right, there is no logic, but no $S^1$-action either :)
– Anton PetruninJun 19 '11 at 13:25

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Compact $G_2$ manifolds also have finite symmetry groups, by the same argument as for the other holonomy groups: a vector field preserving the Riemannian metric would have to be dual to a harmonic 1-form. By Bochner, a harmonic 1-form must be parallel, and so the holonomy reduces, by deRham splitting theorem, to a product of holonomy groups, and the manifold is, up to finite covering, a product with a product metric.
– Ben McKayJun 20 '11 at 8:19

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@Igor: In fact, the holonomy of any metric on the $n$-sphere is $SO(n)$, not just the Ricci-flat ones; i.e., the $n$-sphere does not admit any metric of reduced holonomy.
– Robert BryantJun 21 '11 at 19:57