  We introduce and study a two-parameter family of symmetry reductions of the
two-dimensional Toda lattice hierarchy, which are characterized by a rational
factorization of the Lax operator into a product of an upper diagonal and the
inverse of a lower diagonal formal difference operator. They subsume and
generalize several classical 1 + 1 integrable hierarchies, such as the bigraded
Toda hierarchy, the Ablowitz-Ladik hierarchy and E. Frenkel's q-deformed
Gelfand-Dickey hierarchy. We establish their characterization in terms of block
Toeplitz matrices for the associated factorization problem, and study their
Hamiltonian structure. At the dispersionless level, we show how the
Takasaki-Takebe classical limit gives rise to a family of non-conformal
Frobenius manifolds with flat identity. We use this to generalize the relation
of the Ablowitz-Ladik hierarchy to Gromov-Witten theory by proving an analogous
mirror theorem for rational reductions: in particular, we show that the
dual-type Frobenius manifolds we obtain are isomorphic to the equivariant
quantum cohomology of a family of toric Calabi-Yau threefolds obtained from
minimal resolutions of the local orbifold line.
