  For the two-dimensional Schr\"odinger equation $$ [- \Delta
+v(x)]\psi=E\psi,\ x\in \R^2,\ E=E_{fixed}>0 \ \ \ \ \ (*)$$ at a fixed
positive energy with a fast decaying at infinity potential $v(x)$ dispersion
relations on the scattering data are given.Under "small norm" assumption using
these dispersion relations we give (without a complete proof of sufficiency) a
characterization of scattering data for the potentials from the Schwartz class
$S=C_{\infty}^{(\infty)} (\hbox{\bf R}^2).$ For the potentials with zero
scattering amplitude at a fixed energy $\scriptstyle E_{fixed}$ (transparent
potentials) we give a complete proof of this characterization. As a consequence
we construct a family (parameterized by a function of one variable) of
two-dimensional spherically-symmetric real potentials from the Schwartz class
$S$ transparent at a given energy. For the two-dimensional case (without
assumption that the potential is small) we show that there are no nonzero real
exponentially decreasing at infinity, potentials transparent at a fixed energy.
For any dimension greater or equal 1 we prove that there are no nonzero real
potentials with zero forward scattering amplitude at an energy interval. We
show that KdV-type equations in dimension 2+1 related with the scattering
problem $(*)$ (the Novikov-Veselov equations) do not preserve, in general,
these dispersion relations starting from the second one. As a corollary these
equations do not preserve, in general , the decay rate faster then $|x|^{-3}$
for initial data from the Schwartz class.
