  Ultra-deep observations of ECDF-S with Chandra and XMM-Newton enable a search
for extended X-ray emission down to an unprecedented flux of $2\times10^{-16}$
ergs s$^{-1}$ cm$^{-2}$. We present the search for the extended emission on
spatial scales of 32$^{\prime\prime}$ in both Chandra and XMM data, covering
0.3 square degrees and model the extended emission on scales of arcminutes. We
present a catalog of 46 spectroscopically identified groups, reaching a
redshift of 1.6. We show that the statistical properties of ECDF-S, such as
logN-logS and X-ray luminosity function are broadly consistent with LCDM, with
the exception that dn/dz/d$\Omega$ test reveals that a redshift range of
$0.2<z<0.5$ in ECDF-S is sparsely populated. The lack of nearby structure,
however, makes studies of high-redshift groups particularly easier both in
X-rays and lensing, due to a lower level of clustered foreground. We present
one and two point statistics of the galaxy groups as well as weak-lensing
analysis to show that the detected low-luminosity systems are indeed low-mass
systems. We verify the applicability of the scaling relations between the X-ray
luminosity and the total mass of the group, derived for the COSMOS survey to
lower masses and higher redshifts probed by ECDF-S by means of stacked weak
lensing and clustering analysis, constraining any possible departures to be
within 30% in mass. Abridged.
