  We present observations with the adaptive optics assisted integral field
spectrograph OASIS of the M 31 double nucleus at a spatial resolution better
than 0.5 arcsec FWHM. These data are used to derive the two-dimensional stellar
kinematics within the central 2 arcsec. Archival WFPC2/HST images are revisited
to perform a photometric decomposition of the nuclear region. We also present
STIS/HST kinematics obtained from the archive. The luminosity distribution of
the central region is well separated into the respective contributions of the
bulge, the nucleus including P1 and P2, and the so-called UV peak. We then show
that the axis joining P1 and P2, the two local surface brightness maxima, does
not coincide with the kinematic major-axis, which is also the major-axis of the
nuclear isophotes (excluding P1). We also confirm that the velocity dispersion
peak is offset by ~ 0.2 arcsec from the UV peak, assumed to mark the location
of the supermassive black hole. The newly reduced STIS/HST velocity and
dispersion profiles are then compared to OASIS and other published kinematics.
We find significant offsets with previously published data. Simple parametric
models are then built to successfully reconcile all the available kinematics.
We finally interpret the observations using new N-body simulations. The nearly
keplerian nuclear disk of M31 is subject to a natural m=1 mode, with a very
slow pattern speed (3 km/s/pc for M_BH = 7 10^7~\Msun), that can be maintained
during more than a thousand dynamical times. The resulting morphology and
kinematics of the mode can reproduce the M~31 nuclear-disk photometry and mean
stellar velocity, including the observed asymmetries. It requires a central
mass concentration and a cold disk system representing between 20 and 40% of
its mass. Abridged..
