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 $1-3times10^{-18}$ 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$^{-1}$ ($3sigma$) for two absorbers, and $< 11$~M$_{odot}$ yr$^{-1}$ ($3sigma$) 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.
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