Given a prescribed affine manifold with singularities $(B,\Delta_{\tau},\mathscr{A})$ either as in Example 3.9 in the generic case or as in Example 3.11 in the positive case, there are infinitely many non symplectically conjugate germs of Lagrangian fibrations whose bases are locally affine isomorphic to $(B,\Delta_{\tau},\mathscr{A})$.