  We present the kinematics and photometry of the stars and of the ionized gas
near the centre of the S0 galaxy NGC4036. Dynamical models based on the Jeans
Equation have been constructed from the stellar data to determine the
gravitational potential in which the ionized gas is expected to orbit. Inside
10'', the observed gas rotation curve falls well short of the predicted
circular velocity. Over a comparable radial region the observed gas velocity
dispersion is far higher than the one expected from thermal motions or small
scale turbulence, corroborating that the gas cannot be following the
streamlines of nearly closed orbits. We explore several avenues to understand
the dynamical state of the gas: (1) We treat the gas as a collisionless
ensemble of cloudlets and apply the Jeans Equation to it; this modeling shows
that inside 4'' the gas velocity dispersion is just high enough to explain
quantitatively the absence of rotation. (2) Alternatively, we explore whether
the gas may arise from the `just shed' mass-loss envelopes of the bulge stars,
in which case their kinematics should simply mimic that of the stars. he latter
approach matches the data better than (1), but still fails to explain the low
velocity dispersion and slow rotation velocity of the gas for 5''<r<10''. (3)
Finally, we explore, whether drag forces on the ionized gas may aid in
explaining its peculiar kinematics. While all these approaches provide a much
better description of the data than cold gas on closed orbits, we do not yet
have a definitive model to describe the observed gas kinematics at all radii.
