Finite genus solutions for the Ablowitz-Ladik hierarchy The question of constructing the finite genus quasiperiodic solutions for the Ablowitz-Ladik hierarchy (ALH) is considered by establishing relations between the ALH and the Fay's identity for the theta-functions. It is shown that using a limiting procedure one can derive from the latter an infinite number of differential identities which can be arranged as an infinite set of differential-difference equations coinciding with the equations o...