A The geometry of SU ( 3 ) 51 B The Atiyah - Hitchin system and

In this paper, we look for metrics of cohomogeneity one in D = 8 and D = 7 dimensions with Spin(7) and G 2 holonomy respectively. In D = 8, we first consider the case of principal orbits that are S 7 , viewed as an S 3 bundle over S 4 with triaxial squashing of the S 3 fibres. This gives a more general system of first-order equations for Spin(7) holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over CP 3 , with a non-trivial parameter that characterises the homogeneous squashing of CP 3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N (k, ℓ) = SU (3)/U (1), where the integers k and ℓ characterise the embedding of U (1). We find new ALC and AC metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N (k, ℓ), and numerical and perturbative results for ALC families with AC limits. We then study D = 7 metrics of G 2 holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU (3)/(U (1) × U (1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S 3 × S 3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N (k, ℓ), and an explicit parameterisation of SU (3).