Special cohomogeneity-one metrics with Q1,1,1Q1,1,1 or M1,1,0M1,1,0 as the principal orbit

We classify all cohomogeneity-one manifolds with principal orbit Q1,1,1=SU(2)3/U(1)2Q1,1,1=SU(2)3/U(1)2 or M1,1,0=(SU(3)×SU(2))/(SU(2)×U(1))M1,1,0=(SU(3)×SU(2))/(SU(2)×U(1)) whose holonomy is contained in Spin(7). Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically SU(4)SU(4) and that they are asymptotically conical. Moreover, we investigate the smoothness of the metrics at the singular orbit.