  We show that for the S-state $\pi^0$-production in processes $\gamma+d\to
d+\pi^0$ and $e^-+d\to e^-+d+\pi^0$ the rescattering effects due to the
transition: $ \gamma+d\to p+p+\pi^-$ (or $n+n+\pi^+)\to d+\pi^0$ are cancelled
out due to the Pauli principle. The large values for these effects predicted in
the past may result from the fact that the spin structure of the corresponding
matrix element and the necessary antisymmetrization induced by the presence of
identical protons (or neutrons) in the intermediate state was not taken into
account accurately. One of the important consequences of these considerations
is that $\pi^0$ photo- and electro-production on deuteron near threshold can
bring direct information about elementary neutron amplitudes.
