  The method of nonlinearization of the Lax pair is developed for the
Ablowitz-Kaup-Newell-Segur (AKNS) equation in the presence of space-inverse
reductions. As a result, we obtain a new type of finite-dimensional Hamiltonian
systems: they are nonlocal in the sense that the inverse of the space variable
is involved. For such nonlocal Hamiltonian systems, we show that they preserve
the Liouville integrability and they can be linearized on the Jacobi variety.
We also show how to construct the algebro-geometric solutions to the AKNS
equation with space-inverse reductions by virtue of our nonlocal
finite-dimensional Hamiltonian systems. As an application, algebro-geometric
solutions to the AKNS equation with the Dirichlet and with the Neumann boundary
conditions, and algebro-geometric solutions to the nonlocal nonlinear
Schr\"{o}dinger (NLS) equation are obtained.
