  In this paper we continue our study on the Cauchy problem for the
two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse
scattering transform for the two dimensional Schr\"odinger operator at a fixed
energy parameter. This work is concerned with the case of positive energy. For
the solution of the linearized equation we derive smoothing and Strichartz
estimates by combining two different frequency regimes. At non-low frequencies
we also derive improved smoothing estimates with gain of almost one derivative.
We combine the linear estimates with the Fourier decomposition method and $
X^{s,b} $ spaces to obtain local well-posedness of NV at positive energy in
$H^s$, $s>\frac12$. Our result implies, in particular, that {\it at least} for
$s>\frac12$, NV does not change its behavior from semilinear to quasilinear as
energy changes sign, in contrast to the closely related Kadomtsev-Petviashvili
equations. As a supplement, we provide some new explicit solutions of NV at
zero energy which exhibit an interesting behavior at large times.
