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. We outline observational tests to understand the enigmatic nature of the ionized gas.