  We exploit the increased sensitivity of the recently installed AO SOUL at the
LBT to obtain new high-spatial-resolution NIR images of the massive young
stellar object IRAS20126+4104 and its outflow. We aim to derive the jet proper
motions and kinematics, as well as to study its photometric variability by
combining the novel performances of SOUL together with previous NIR images. We
used both broad-band ($K_{s}$, $K'$) and narrow-band (Br$\gamma$, H2)
observations from a number of NIR cameras
(UKIRT/UFTI,SUBARU/CIAO,TNG/NICS,LBT/PISCES,and LBT/LUCI1) to derive maps of
the continuum and the H$_2$ emission in the 2.12 $\mu$m line. Three sets of
images, obtained with AO systems (CIAO,2003; FLAO,2012; SOUL,2020), allowed us
to derive the proper motions of a large number of H$_2$ knots along the jet.
Photometry from all images was used to study the jet variability. We derived
knot proper motions in the range of 1.7-20.3 mas yr$^{-1}$ (i.e. 13-158 km
s$^{-1}$ at 1.64 kpc, avg. outflow tangential velocity $\sim$ 80 km s$^{-1}$).
The derived knot dynamical age spans a $\sim$ 200-4000 yr interval. A ring-like
H$_2$ feature near the protostar location exhibits peculiar kinematics and may
represent the outcome of a wide-angle wind impinging on the outflow cavity.
Both H$_2$ geometry and velocities agree with those inferred from proper
motions of the H$_2$O masers, located at a smaller distance from the protostar.
Although the total H$_2$ line emission from the knots does not exhibit time
variations at a $\widetilde{>}$ 0.3 mag level, we have found a clear continuum
flux variation (radiation scattered by the dust in the cavity opened by the
jet) which is anti-correlated between the blue-shifted and red-shifted lobes
and may be periodic (with a period of $\sim$ 12-18 yr). We suggest that the
continuum variability might be related to inner-disc oscillations which have
also caused the jet precession.
