High resolution imaging and spectroscopy of the gravitational lens
  SDSSJ1206+4332: a natural coronagraph at $z=1.789$ and a standard ruler at
  $z=0.745$
  We present spectroscopy and laser guide star adaptive optics (LGSAO) images
of the doubly imaged lensed quasar SDSS J1206+4332. We revise the deflector
redshift proposed previously to $z_{d}=0.745,$ and measure for the first time
its velocity dispersion $\sigma=(290\pm30)$ km/s. The LGSAO data show the
lensed quasar host galaxy stretching over the astroid caustic thus forming an
extra pair of merging images, which was previously thought to be an unrelated
galaxy in seeing limited data. Owing to the peculiar geometry, the lens acts as
a natural coronagraph on the broad-line region of the quasar so that only
narrow [O III] emission is found in the fold arc. We use the data to
reconstruct the source structure and deflector potential, including nearby
perturbers. We reconstruct the point-spread function (PSF) from the quasar
images themselves, since no additional point source is present in the field of
view. From gravitational lensing and stellar dynamics, we find the slope of the
total mass density profile to be $\gamma^{\prime}=-\log\rho/\log r
=1.93\pm0.09.$ We discuss the potential of SDSS J1206+4332 for measuring time
delay distance (and thus H$_0$ and other cosmological parameters), or as a
standard ruler, in combination with the time delay published by the COSMOGRAIL
collaboration. We conclude that this system is very promising for cosmography.
However, in order to achieve competitive precision and accuracy, an independent
characterization of the PSF is needed. Spatially resolved kinematics of the
deflector would reduce the uncertainties further. Both are within the reach of
current observational facilities.
