  Based on the completeness relation for the squared solutions of the Lax
operator $L$ we show that a subset of nonlocal equations from the hierarchy of
nonlocal nonlinear Schr\"odinger equations (NLS) is a completely integrable
system. The spectral properties of the Lax operator indicate that there are two
types of soliton solutions. The relevant action-angle variables are
parametrized by the scattering data of the Lax operator. The notion of the
symplectic basis, which directly maps the variations of the potential of $L$ to
the variations of the action-angle variables has been generalized to the
nonlocal case. We also show that the inverse scattering method can be viewed as
a generalized Fourier transform. Using the trace identities and the symplectic
basis we construct the hierarchy Hamiltonian structures for the nonlocal NLS
equations.
