  We study the production of the lightest neutralinos in the radiative process
$e^+e^- \to \tilde\chi^0_1 \tilde\chi^0_1\gamma$ in low energy supersymmetric
models for the International Linear Collider energies with longitudinally
polarized electron and positron beams. For this purpose we consider the case of
nonminimal supersymmetric standard model as well as the case of minimal
supersymmetric standard model. At the first stage of a linear collider, with
$\sqrt{s} =500$ GeV, the radiative production of the lightest neutralinos may
be a viable channel to study supersymmetric partners of the Standard Model
particles, especially if the other supersymmetric particles are too heavy to to
be pair-produced. We consider in detail the effect of beam polarization on the
production cross section. We compare and contrast the dependence of the signal
cross section on the parameters of the neutralino sector of the nonminimal and
minimal supersymmetric standard model when the electron and positron beams are
longitudinally polarized. In order to assess the feasibility of experimentally
observing the radiative neutralino production process, we consider the
background to this process coming from the Standard Model process $e^+e^- \to
\nu \bar\nu \gamma$ with longitudinally polarized electron and positron beams.
We also consider the supersymmetric background to the radiative neutralino
production process coming from the radiative production of the scalar partners
of the neutrinos (sneutrinos) $e^+e^- \to \tilde\nu \tilde\nu^\ast \gamma$,
with longitudinally polarized beams. This process can be a a background to the
radiative neutralino production when the sneutrinos decay invisibly.
