  There are about 33, 6.5 and 1.0 pb$^{-1}$ of $e^+e^-$ annihilation data have
been taken around the center-of-mass energies of $\sqrt s=$ 3.773 GeV, at
$\sqrt s=$ 3.650 GeV and at $\sqrt s=$ 3.6648 GeV, respectively, with the
BES-II detector at the BEPC collider. By analyzing these data sets, we have
measured the branching fraction for $\psi(3770)\to$ non-$D\bar D$ by several
different methods; and have observed an anomalous line shape of
$\sigma^{e^+e^-\to{\rm hadrons}}$ in energy region from 3.65 to 3.87 GeV; and
have measured the line shapes of the $D^+D^-$, $D^0\bar D^0$ and $D\bar D$
production and together with the ratios of the production rates of $D^+D^-$ and
$D^0\bar D^0$ in $e^+e^-$ annihilation around the $\psi(3770)$ resonance.
