Lagrangian fibrations on Hilbert schemes of points on K3 surfaces

Let ${\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^{g}S$ be the Hilbert scheme of $g$ points on a K3 surface $S$. Suppose that $\mathrm{P}\mathrm{i}\mathrm{c}S\cong \mathbb{Z}C$ where $C$ is a smooth curve with $C^2=2(g-1)n^2$. We prove that ${\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^{g}S$ is a Lagrangian fibration.