  We study Ricci-flat metrics on non-compact manifolds with the exceptional
holonomy $Spin(7), G_2$. We concentrate on the metrics which are defined on
${\bf R} \times G/H$. If the homogeneous coset spaces $G/H$ have weak $G_2$,
SU(3) holonomy, the manifold ${\bf R} \times G/H$ may have $Spin(7), G_2$
holonomy metrics. Using the formulation with vector fields, we investigate the
metrics with $Spin(7)$ holonomy on ${\bf R}\times Sp(2)/Sp(1), {\bf R}\times
SU(3)/U(1)$. We have found the explicit volume-preserving vector fields on
these manifold using the elementary coordinate parameterization. This
construction is essentially dual to solving the generalized self-duality
condition for spin connections. We present most general differential equations
for each coset. Then, we develop the similar formulation in order to calculate
metrics with $G_2$ holonomy
