  We report on a sensitive search for redshifted H$\alpha$ line-emission from
three high-metallicity damped Ly$\alpha$ absorbers (DLAs) at $z \approx 2.4$
with the Near-infrared Integral Field Spectrometer (NIFS) on the Gemini-North
telescope, assisted by the ALTtitude conjugate Adaptive optics for the InfraRed
(ALTAIR) system with a laser guide star. Within the NIFS field-of-view,
$\approx 3.22" \times 2.92"$ corresponding to $\approx 25$ kpc $ \times 23$ kpc
at $z=2.4$, we detect no statistically significant line-emission at the
expected redshifted H$\alpha$ wavelengths. The measured root-mean-square noise
fluctuations in $0.4"$ apertures are $1-3\times10^{-18}$ erg s$^{-1}$
cm$^{-2}$. Our analysis of simulated, compact, line-emitting sources yields
stringent limits on the star-formation rates (SFRs) of the three DLAs, $<
2.2$~M$_{\odot}$ yr$^{-1}$ ($3\sigma$) for two absorbers, and $<
11$~M$_{\odot}$ yr$^{-1}$ ($3\sigma$) for the third, at all impact parameters
within $\approx 12.5$~kpc to the quasar sightline at the DLA redshift. For the
third absorber, the SFR limit is $< 4.4$~M$_\odot$ yr$^{-1}$ for locations away
from the quasar sightline. These results demonstrate the potential of adaptive
optics-assisted, integral field unit searches for galaxies associated with
high-$z$ DLAs.
