We present a determination of the mass of the supermassive black hole (BH) and the nuclear stellar orbital distribution of the elliptical galaxy Centaurus A (NGC5128) using high-resolution integral-field observations of the stellar kinematics. The observations were obtained with SINFONI at the ESO Very Large Telescope in the near-infrared (K-band), using adaptive optics to correct for the blurring effect of the earth atmosphere. The data have a spatial resolution of 0.17 FWHM and high S/N>80 per spectral pixel so that the shape of the stellar line-of-sight velocity-distribution can be reliably extracted. We detect clear low-level stellar rotation, which is counter-rotating with respect to the gas. We fit axisymmetric three-integral dynamical models to the data to determine the best fitting values for the BH mass M_BH=(5.5+/-3.0)*10^7 Msun (3sigma errors) and (M/L)_K=(0.65+/-0.15) in solar units. These values are in excellent agreement with previous determinations from the gas kinematics, and in particular with our own published values, extracted from the same data. This provides one of the cleanest gas versus stars comparisons of BH determination, due to the use of integral-field data for both dynamical tracers and due to a very well resolved BH sphere of influence R_BH~0.70. We derive an accurate profile of the orbital anisotropy and we carefully test its reliability using spherical Jeans models with radially varying anisotropy. We find an increase in the tangential anisotropy close to the BH, but the spatial extent of this effect seems restricted to the size of R_BH instead of that R_b~3.9 of the core in the surface brightness profile, contrary to detailed predictions of current simulations of the binary BH scouring mechanism. More realistic simulations would be required to draw conclusions from this observation.
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