Let $(B,\Delta,\mathscr{A})$ be the affine manifold with singularities in Example 3.16 and let $X(B_{0},\mathscr{A})$ be the associated $T^{2}$ bundle with symplectic structure $\omega_{0}$ and projection $f_{0}$ induced by the standard ones in $T_{B_{0}}^{\ast}$. There exists a compact symplectic manifold $(X,\omega)$, a Lagrangian fibration $f:X\rightarrow B$ and an embedding $\iota:X(B_{0},\mathscr{A})\rightarrow X$ such that $\iota^{\ast}\omega=\omega_{0}$ and $f\circ\iota=f_{0}$. Moreover $X$ is diffeomorphic to a smooth $K3$ surface.