2 Einstein equation and first integrals for Spin ( 7 ) metrics

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A 8 , is complete and non-singular on R 8. The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S 4 , and are denoted by B + 8 , B − 8 and B 8. The metrics on B + 8 and B − 8 occur in families with a non-trivial parameter. The metric on B 8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A 8. The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP 3. We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L 2-normalisable harmonic 4-form for the A 8 manifold, and two such 4-forms (of opposite dualities) for the B 8 manifold.