  For the first time we search for the $\eta_{c2}(1D)$ in
$e^+e^-\to\gamma\eta_{c2}(1D)$ at $\sqrt{s}$ = 10.52, 10.58, and 10.867 GeV
with data samples of 89.5 fb$^{-1}$, 711 fb$^{-1}$, and 121.4 fb$^{-1}$,
respectively, accumulated with the Belle detector at the KEKB asymmetric energy
electron-positron collider. No significant $\eta_{c2}(1D)$ signal is observed
in the mass range between 3.8 and 3.88 GeV/$c^2$. The upper limit at 90\%
confidence level on the product of the Born cross section for $e^+e^- \to
\gamma\eta_{c2}(1D)$ and branching fraction for $\eta_{c2}(1D)\to \gamma
h_c(1P)$ is determined to be $\sigma(e^+e^- \to \gamma\eta_{c2}(1D)){\cal
B}(\eta_{c2}(1D)\to \gamma h_c(1P))$ $<$ 4.9 fb at $\sqrt{s}$ near 10.6 GeV.
