Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold

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 $\mathcal{O}_{CP^1} (-1)$.