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In this paper, we show that there is a natural Lagrangian fibration structureon the map $\Phi$ from the cotangent bundle of a del Pezzo surface $X$ ofdegree 4 to $\mathbb C^2$. Moreover, we describe explicitly all level surfacesof the above natural map $\Phi$.

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