  We continue the investigation of Spin(7) holonomy metric of cohomogeneity one
with the principal orbit SU(3)/U(1). A special choice of U(1) embedding in
SU(3) allows more general metric ansatz with five metric functions. There are
two possible singular orbits in the first order system of Spin(7) instanton
equation. One is the flag manifold $SU(3)/T^2$ also known as the twister space
of CP(2) and the other is CP(2) itself. Imposing a set of algebraic
constraints, we find a two-parameter family of exact solutions which have SU(4)
holonomy and are asymptotically conical. There are two types of asymptotically
locally conical (ALC) metrics in our model, which are distingushed by the
choice of $S^1$ circle whose radius stabilizes at infinity. We show that this
choice of M theory circle selects one of possible singular orbits mentioned
above. Numerical analyses of solutions near the singular orbit and in the
asymptotic region support the existence of two families of ALC Spin(7) metrics:
one family consists of deformations of the Calabi hyperKahler metric, the other
is a new family of metrics on a line bundle over the twister space of CP(2).
