  Using a method introduced by Hitchin we obtain the system of first order
differential equations that determine the most general cohomogeniety one G_2
holonomy metric with S^3 \times S^3 principal orbits. The method is then
applied to G_2 metric with S^3 \times T^3 principal orbits in which an analytic
solution is obtained. The generalized metric has more free parameters than that
previously constructed. After showing that the generalization is non-trivial a
system of first order equations is obtained for new Spin(7) metric with
principal orbits S^7.
