  We give a general description of the construction of weighted spherically
symmetric metrics on vector bundle manifolds, i.e. the total space of a vector
bundle $E\rightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed
with a metric connection. The tangent bundle of $E$ admits a canonical
decomposition and thus it is possible to define an interesting class of
two-weights metrics with the weight functions depending on the fibre norm of
$E$; hence the generalized concept of spherically symmetric metrics. We study
its main properties and curvature equations. Finally we focus on a few
applications and compute the holonomy of Bryant-Salamon type $\mathrm{G}_2$
manifolds.
