Integrable hierarchies and the mirror model of local CP1

► Structural aspects of Ablowitz-Ladik hierarchy as a rational reduction of 2D Toda. ► Non-exact bi-Hamiltonian structures of hydrodynamic type and Frobenius manifolds with non-flat unit. ► Mirror symmetry for Gromov-Witten theory of local CP1. ► The mirror model to local CP1 is almost dual to the Ablowitz-Ladik Frobenius manifold.