  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 of the ALH, and
that the original Fay's identity can be rewritten in a form similar to the
functional equations representing the ALH which have been derived in the
previous works of the author. This provides an algorithm for obtaining some
class of quasiperiodic solutions for the ALH, which can be viewed as an
alternative to the inverse scattering transform or the algebro-geometrical
approach.
