We study the stability of stellar dynamical equilibrium models for M32 .
Kinematic observations show that M32 has a central dark mass of \sim 3 \times 10 ^ { 6 } \ > { M _ { \odot } } , most likely a black hole , and a phase-space distribution function that is close to the ‘ two-integral ’ form f = f ( E,L _ { z } ) .
M32 is also rapidly rotating ; 85–90 % of the stars have the same sense of rotation around the symmetry axis .
Previous work has shown that flattened , rapidly rotating two-integral models can be bar-unstable .
We have performed N-body simulations to test whether this is the case for M32 .
This is the first stability analysis of two-integral models that have both a central density cusp and a nuclear black hole .

Particle realizations with N = \ > 512,000 were generated from distribution functions that fit the photometric and kinematic data of M32 .
We constructed equal-mass particle realizations , and also realizations with a mass spectrum to improve the central resolution .
Models were studied for two representative inclinations , i = 90 ^ { \circ } ( edge-on ) and i = 55 ^ { \circ } , corresponding to intrinsic axial ratios of q = 0.73 and q = 0.55 , respectively .
The time evolution of the models was calculated with a ‘ self-consistent field ’ code on a Cray T3D parallel supercomputer .
We find both models to be dynamically stable .
This implies that they provide a physically meaningful description of M32 , and that the inclination of M32 ( and hence its intrinsic flattening ) can not be strongly constrained through stability arguments .

Previous work on the stability of f ( E,L _ { z } ) models has shown that the bar-mode is the only possibly unstable mode for systems rounder than q \approx 0.3 ( i.e. , E7 ) , and that the likelihood for this mode to be unstable increases with flattening and rotation rate .
The f ( E,L _ { z } ) models studied for M32 are stable , and M32 has a higher rotation rate than nearly all other elliptical galaxies .
This suggests that f ( E,L _ { z } ) models constructed to fit data for real elliptical galaxies will generally be stable , at least for systems rounder than q \gtrsim 0.55 , and possibly for flatter systems as well .