  Coassociative submanifolds are 4-dimensional calibrated submanifolds in
$G_{2}$-manifolds. In this paper, we construct explicit examples of
coassociative submanifolds in $\Lambda^{2}_{-} S^{4}$, which is the complete
$G_{2}$-manifold constructed by Bryant and Salamon. Classifying the Lie groups
which have 3- or 4-dimensional orbits, we show that the only homogeneous
coassociative submanifold is the zero section of $\Lambda^{2}_{-} S^{4}$ up to
the automorphisms and construct many cohomogeneity one examples explicitly. In
particular, we obtain examples of non-compact coassociative submanifolds with
conical singularities and their desingularizations.
