  On each complete asymptotically conical $Spin(7)$ manifold constructed by
Bryant and Salamon, including the asymptotic cones, we consider a natural
family of $SU(2)$ actions preserving the Cayley form. For each element of this
family, we study the (possibly singular) invariant Cayley fibration, which we
describe explicitly, if possible. These can be reckoned as generalizations of
the trivial flat fibration of $\mathbb{R}^8$ and the product of a line with the
Harvey-Lawson coassociative fibration of $\mathbb{R}^7$. The fibres will
provide new examples of asymptotically conical Cayley submanifolds in the
Bryant-Salamon manifolds of topology $\mathbb{R}^4, \mathbb{R}\times S^3$ and
$\mathcal{O}_{CP^1} (-1)$.
