  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. This includes the
minimal supersymmetric standard model as well as its extension with an
additional chiral Higgs singlet superfield, the nonminimal supersymmetric
standard model. We compare and contrast the dependence of the signal cross
section on the parameters of the neutralino sector of the minimal and
nonminimal supersymmetric standard model. We also consider the background to
this process coming from the Standard Model process $e^+e^- \to \nu \bar\nu
\gamma$, as well as from the radiative production of the scalar partners of the
neutrinos (sneutrinos) $e^+e^- \to \tilde\nu \tilde\nu^\ast \gamma$, which can
be a background to the radiative neutralino production when the sneutrinos
decay invisibly. In low energy supersymmetric models radiative production of
the lightest neutralinos may be the only channel to study supersymmetric
partners of the Standard Model particles at the first stage of a linear
collider, since heavier neutralinos, charginos and sleptons may be too heavy to
be pair-produced at a $e^+ e^-$ machine with $\sqrt{s} =500\GeV$.
