  We present an analysis of the H2 emission-line gas kinematics in the inner <
4 arcsec radius of six nearby spiral galaxies, based on AO-assisted
integral-field observations obtained in the K-band with SINFONI/VLT. Four of
the six galaxies in our sample display ordered H2 velocity fields, consistent
with gas moving in the plane of the galaxy and rotating in the same direction
as the stars. However, the gas kinematics is typically far from simple circular
motion. We can classify the observed velocity fields into four different types
of flows, ordered by increasing complexity: (1) circular motion in a disc
(NGC3351); (2) oval motion in the galaxy plane (NGC3627 and NGC4536); (3)
streaming motion superimposed on circular rotation (NGC4501); and (4)
disordered streaming motions (NGC4569 and NGC4579). The H2 velocity dispersion
in the galaxies is usually higher than 50 km/s in the inner 1-2 arcsec radii.
The four galaxies with ordered kinematics have v/sigma < 1 at radii less than
40-80 pc. The radius at which v/sigma = 1 is independent of the type of nuclear
activity. While the low values of v/sigma could be taken as an indication of a
thick disc in the innermost regions of the galaxies, other lines of evidence
(e.g. H2 morphologies and velocity fields) argue for a thin disc interpretation
in the case of NGC3351 and NGC4536. We discuss the implications of the high
values of velocity dispersion for the dynamics of the gaseous disc and suggest
caution when interpreting the velocity dispersion of ionized and warm tracers
as being entirely dynamical. Understanding the nature and role of the velocity
dispersion in the gas dynamics, together with the full 2D information of the
gas, is essential for obtaining accurate black hole masses from gas kinematics.
