  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]
