  A class of non-local non-linear Schrodinger equations(NLSE) is considered in
an external potential with space-time modulated coefficient of the nonlinear
interaction term as well as confining and/or loss-gain terms. This is a
generalization of a recently introduced integrable non-local NLSE with
self-induced potential that is PT symmetric in the corresponding stationary
problem. Exact soliton solutions are obtained for the inhomogeneous and/or
non-autonomous non-local NLSE by using similarity transformation and the method
is illustrated with a few examples. It is found that only those transformations
are allowed for which the transformed spatial coordinate is odd under the
parity transformation of the original one. It is shown that the non-local NLSE
without the external potential and a d+1 dimensional generalization of it,
admits all the symmetries of the d+1 dimensional Schrodinger group. The
conserved Noether charges associated with the time-translation, dilatation and
special conformal transformation are shown to be real-valued in spite of being
non-hermitian. Finally, dynamics of different moments are studied with an exact
description of the time-evolution of the "pseudo-width" of the wave-packet for
the special case when the system admits a O(2,1) conformal symmetry.
