We present results from a search for strong H2 absorption systems proximate to quasars (zabs~zem) in the Sloan Digital Sky Survey (SDSS) Data Release 14. The search is based on the Lyman-Werner band signature of damped H2 absorption lines without any prior on the associated metal or neutral hydrogen content. This has resulted in the detection of 81 systems with log N(H2)~19-20 located within a few thousand km/s from the quasar. Compared to a control sample of intervening systems, this implies an excess of proximate H2 systems by about a factor of 4 to 5. The incidence of H2 systems increases steeply with decreasing relative velocity, reaching an order of magnitude higher than expected from intervening statistics at Delta_v<1000 km/s. The most striking feature of the proximate systems compared to the intervening ones is the presence of Ly-alpha emission in the core of the associated damped HI absorption line in about half of the sample. This puts constraints on the relative projected sizes of the absorbing clouds to those of the quasar line emitting regions. Using the SDSS spectra, we estimate the HI, metal and dust content of the systems, which are found to have typical metallicities of one tenth Solar, albeit with a large spread among individual systems. We observe trends between the fraction of leaking Ly-alpha emission and the relative absorber-quasar velocity as well as with the excitation of several metal species, similar to what has been seen in metal-selected proximate DLAs. With the help of theoretical HI-H2 transition relations, we show that the presence of H2 helps to break the degeneracy between density and strength of the UV field as main sources of excitation and hence provides unique constraints on the possible origin and location of the absorbing clouds. We suggest that most of these systems originate from galaxies in the quasar group. [truncated]
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