  Rational curves on Hilbert schemes of points on $K3$ surfaces and generalised
Kummer manifolds are constructed by using Brill-Noether theory on nodal curves
on the underlying surface. It turns out that all wall divisors can be obtained,
up to isometry, as dual divisors to such rational curves. The locus covered by
the rational curves is then described, thus exhibiting algebraically
coisotropic subvarieties that deform to general small deformations of the
manifold. This provides strong evidence for a conjecture by Voisin concerning
the Chow ring of irreducible holomorphic symplectic manifolds. Some general
results concerning the birational geometry of irreducible holomorphic
symplectic manifolds are also proved, such as a non-projective contractibility
criterion for wall divisors.
