  The aim of this paper is to develop the inverse scattering transform (IST)
for multi-component generalisations of nonlocal reductions of the nonlinear
Schrodinger (NLS) equation with PT-symmetry related to symmetric spaces. This
includes: the spectral properties of the associated Lax operator, Jost
function, the scattering matrix and the minimal set of scattering data, the
fundamental analytic solutions. As main examples, we use the Manakov vector
Schr\"odinger equation (related to A.III-symmetric spaces) and the
multi-component NLS (MNLS) equations of Kullish-Sklyanin type (related to
BD.I-symmetric spaces). Furthermore, the 1- and 2-soliton solutions are
obtained by using an appropriate modification of the Zakharov-Shabat dressing
method. It is shown, that the MNLS equations of these types allow both regular
and singular soliton configurations. Finally, we present here different
examples of 1- and 2-soliton solutions for both types of models, subject to
different reductions.
