  We study families of rational curves on irreducible holomorphic symplectic
varieties. We give a necessary and sufficient condition for a sufficiently
ample linear system on a holomorphic symplectic variety of $K3^{[n]}$-type to
contain a uniruled divisor covered by rational curves of primitive class. In
particular, for any fixed $n$, we show that there are only finitely many
polarization types of holomorphic symplectic variety of $K3^{[n]}$-type that do
not contain such a uniruled divisor. As an application we provide a
generalization of a result due to Beauville-Voisin on the Chow group of
0-cycles on such varieties.
