  The inner disc of the local group galaxy M33 appears to be in settled
rotational balance, and near IR images reveal a mild, large-scale, two-arm
spiral pattern with no strong bar. We have constructed N-body models that match
all the extensive observational data on the kinematics and surface density of
stars and gas in the inner part of M33. We find that currently favoured models
are unstable to the formation of a strong bar of semi-major axis 2 < a_B < 3
kpc within 1 Gyr, which changes the dynamical properties of the models to
become inconsistent with the current, apparently well-settled, state. The
formation of a bar is unaffected by how the gas component is modelled, by
increasing the mass of the nuclear star cluster, or by making the dark matter
halo counter-rotate, but it can be prevented by either reducing the
mass-to-light ratio of the stars to Upsilon_V ~ 0.6 or Upsilon_K ~ 0.23 in
solar units or by increasing the random motions of the stars. Also a shorter
and weaker bar results when the halo is rigid and unresponsive. However, all
three near-stable models support multi-arm spirals, and not the observed
large-scale bi-symmetric spiral. A two-arm spiral pattern could perhaps be
tidally induced, but such a model would require an implausibly low mass disc to
avoid a bar and there is no visible culprit. Thus the survival of the current
state of this exceptionally well-studied galaxy is not yet understood. We also
suspect that many other unbarred galaxies present a similar puzzle.
