  A study of exclusive production of the $D \bar D$ system through
initial-state r adiation is performed in a search for charmonium states, where
$D=D^0$ or $D^+$. The $D^0$ mesons are reconstructed in the $D^0 \to K^-
\pi^+$, $D^0 \to K^- \pi^+ \pi^0$, and $D^0 \to K^- \pi^+ \pi^+ \pi^-$ decay
modes. The $D^+$ is reconstructed through the $D^+ \to K^- \pi^+ \pi^+$ decay
mode. The analysis makes use of an integrated luminosity of 288.5 fb$^{-1}$
collected by the BaBar experiment. The $D \bar D$ mass spectrum shows a clear
$\psi(3770)$ signal. Further structures appear in the 3.9 and 4.1 GeV/$c^2$
regions. No evidence is found for Y(4260) decays to $D \bar D$, implying an up
per limit $\frac{\BR(Y(4260)\to D \bar D)}{\BR(Y(4260)\to J/\psi \pi^+ \pi^-)}
< 7.6$ (95 % confidence level).
