Lagrangian fibration structure on the cotangent bundle of a del Pezzo surface of degree 4

In this paper, we show that there is a natural Lagrangian fibration structure on the map $\mathrm{\Phi }$ from the cotangent bundle of a del Pezzo surface $X$ of degree 4 to ${\mathbb{C}}^2$. Moreover, we describe explicitly all level surfaces of the above natural map $\mathrm{\Phi }$.