  The SINFONI survey for Unveiling the Physics and Effect of Radiative feedback
(SUPER) aims at tracing and characterizing ionized gas outflows and their
impact on star formation in a statistical sample of X-ray selected Active
Galactic Nuclei (AGN) at z$\sim$2. We present the first SINFONI results for a
sample of 21 Type-1 AGN spanning a wide range in bolometric luminosity (log
$\mathrm{L_{bol}}$ = 45.4-47.9 erg/s). The main aims of this paper are
determining the extension of the ionized gas, characterizing the occurrence of
AGN-driven outflows, and linking the properties of such outflows with those of
the AGN. We use Adaptive Optics-assisted SINFONI observations to trace ionized
gas in the extended narrow line region using the [OIII]5007 line. We classify a
target as hosting an outflow if its non-parametric velocity of the [OIII] line,
$\mathrm{w_{80}}$, is larger than 600 km/s. We study the presence of extended
emission using dedicated point-spread function (PSF) observations, after
modelling the PSF from the Balmer lines originating from the Broad Line Region.
We detect outflows in all the Type-1 AGN sample based on the $\mathrm{w_{80}}$
value from the integrated spectrum, which is in the range 650-2700 km/s. There
is a clear positive correlation between $\mathrm{w_{80}}$ and the AGN
bolometric luminosity (99% correlation probability), but a weaker correlation
with the black hole mass (80% correlation probability). A comparison of the PSF
and the [OIII] radial profile shows that the [OIII] emission is spatially
resolved for $\sim$35% of the Type-1 sample and the outflows show an extension
up to $\sim$6 kpc. The relation between maximum velocity and the bolometric
luminosity is consistent with model predictions for shocks from an AGN driven
outflow. The escape fraction of the outflowing gas increase with the AGN
luminosity, although for most galaxies, this fraction is less than 10%.
