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We present Keck Cosmic Web Imager (KCWI) Ly$\alpha$ integral fieldspectroscopy of the fields surrounding 14 Damped Ly$\alpha$ absorbers (DLAs) at$z \approx 2$. Of these 14 DLAs, 9 have high metallicities ([M/H]$~> -0.3$),and 4 of those 9 feature a CO-emitting galaxy at an impact parameter $\lesssim30$ kpc. Our search reaches median Ly$\alpha$ line flux sensitivities of $\sim2 \times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ over apertures of $\sim6$ kpc and outto impact parameters of $\sim50$ kpc. We recover the Ly$\alpha$ flux of threeknown Ly$\alpha$-emitting H I-selected galaxies in our sample. In addition, wefind two Ly$\alpha$ emitters at impact parameters of $\approx 50-70$ kpc fromthe high metallicity DLA at $z \approx 1.96$ toward QSO B0551-366. This fieldalso contains a massive CO-emitting galaxy at an impact parameter of $\approx15$ kpc. Apart from the field with QSO B0551-366, we do not detect significantLy$\alpha$ emission in any of the remaining 8 high-metallicity DLA fields.Considering the depth of our observations and our ability to recover previouslyknown Ly$\alpha$ emitters, we conclude that H I-selected galaxies associatedwith high-metallicity DLAs at $z \approx 2$ are dusty, and therefore mightfeature low Ly$\alpha$ escape fractions. Our results indicate thatcomplementary approaches -- using Ly$\alpha$, CO, H$\alpha$, and [C II]158$\mu$m emission -- are necessary to identify the wide range of galaxy typesassociated with $z \approx 2$ DLAs.

A survey of Ly$\alpha$ emission around Damped Ly$\alpha$ absorbers at $z \approx 2$ with the Keck Cosmic Web Imager