In a recent paper, we have pointed out a relation between the Killing spinor and Einstein equations. Using this relation, new brane solutions of D=11 and D=10 type IIB supergravity theories are constructed. It is shown that in a brane solution, the flat world-volume directions, the smeared transverse directions and the sphere located at a fixed radial distance can be replaced with any Lorentzian Ricci flat, Euclidean Ricci flat and Einstein manifolds, respectively. The solution obtained in this fashion is supersymmetric when the Ricci flat and Einstein manifolds have Killing spinors. We generalize intersecting brane solutions, in which M5-, M2- and D3-branes also wrap over the cycles determined by the Kähler forms of Ricci flat Kähler manifolds. New, singular, Ricci flat manifolds as (generalized) cones over the U(1) bundles over Ricci flat Kähler spaces are constructed. These manifolds have covariantly constant spinors and give rise to new, supersymmetric, Ricci flat compactifications of non-gauged supergravity theories. We find M2- and D3-brane solutions, which asymptotically approach these singular vacua.