We have obtained very deep near-infrared images in the fields of 10 QSOs whose spectra contain damped Lyman-alpha absorption (DLA) systems with 1.7<z_abs <2.5. The main aim of our investigation is to provide new constraints on the properties of distant galaxies responsible for the DLA absorption. After subtracting the point spread function associated with the QSO light, we have detected galaxies very close to the QSO line of sight (projected distance 1.2-1.3arcsec) in two fields. These sources therefore represent promising candidate galaxies responsible for the DLA absorption. Placed at the absorbers redshift, the impact parameter is 10h_50^-1kpc and the luminosity is close to L_K^*. Such parameters are consistent with the hypothesis, verified for metallic systems at lower redshift, that slowly-evolving massive galaxies produce at least some of the absorption systems of high column density in QSO spectra out to beyond z=2. In addition to detecting these candidate DLA galaxies, the radio-loud QSOs in our sample show a significant excess of sources on larger scales (theta=7arcsec); this excess is not present in the radio-quiet QSO sightlines. We calculate that such an excess could be produced by luminous galaxies in the cores of clusters associated with radio-loud QSOs. Both results confirm that deep imaging of selected QSOs can be a powerful method of finding samples of likely z~2 galaxies. Follow-up near-infrared spectroscopy is required to secure galaxy redshifts and star formation rates, while deep HST imaging can determine sizes and morphologies, providing valuable information on galaxy properties at large look-back times.
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