  We present spectroscopic observations of six high redshift ($z_{\rm em}$ $>$
2) quasars, which have been selected for their Lyman $\alpha$ (Ly$\alpha$)
emission region being only partially covered by a strong proximate ($z_{\rm
abs}$ $\sim$ $z_{\rm em}$) coronagraphic damped Ly$\alpha$ system (DLA). We
detected spatially extended Ly$\alpha$ emission envelopes surrounding these six
quasars, with projected spatial extent in the range 26 $\le$ $d_{\rm Ly\alpha}$
$\le$ 51 kpc. No correlation is found between the quasar ionizing luminosity
and the Ly$\alpha$ luminosity of their extended envelopes. This could be
related to the limited covering factor of the extended gas and/or due to the
AGN being obscured in other directions than towards the observer. Indeed, we
find a strong correlation between the luminosity of the envelope and its
spatial extent, which suggests that the envelopes are probably ionized by the
AGN. The metallicity of the coronagraphic DLAs is low and varies in the range
$-$1.75 $<$ [Si/H] $<$ $-$0.63. Highly ionized gas is observed to be associated
with most of these DLAs, probably indicating ionization by the central AGN. One
of these DLAs has the highest AlIII/SiII ratio ever reported for any
intervening and/or proximate DLA. Most of these DLAs are redshifted with
respect to the quasar, implying that they might represent infalling gas
probably accreted onto the quasar host galaxies through filaments.
