  We present a dynamical analysis to infer the structural parameters and
properties of the two nearby, compact, high velocity dispersion galaxies
MRK1216 & NGC1277. Combining deep HST imaging, wide-field IFU stellar
kinematics, and complementary long-slit spectroscopic data out to 3 R_e, we
construct orbit-based models to constrain their black hole masses, dark matter
content and stellar mass-to-light ratios. We obtain a black hole mass of
log(Mbh/Msun) = 10.1(+0.1/-0.2) for NGC1277 and an upper limit of log(Mbh/Msun)
= 10.0 for MRK1216, within 99.7 per cent confidence. The stellar mass-to-light
ratios span a range of Upsilon_V = 6.5(+1.5/-1.5) in NGC1277 and Upsilon_H =
1.8(+0.5/-0.8) in MRK1216 and are in good agreement with SSP models of a single
power-law Salpeter IMF. Even though our models do not place strong constraints
on the dark halo parameters, they suggest that dark matter is a necessary
ingredient in MRK1216, with a dark matter contribution of 22(+30/-20) per cent
to the total mass budget within 1 R_e. NGC1277, on the other hand, can be
reproduced without the need for a dark halo, and a maximal dark matter fraction
of 13 per cent within the same radial extent. In addition, we investigate the
orbital structures of both galaxies, which are rotationally supported and
consistent with photometric multi-S\'ersic decompositions, indicating that
these compact objects do not host classical, non-rotating bulges formed during
recent (z <= 2) dissipative events or through violent relaxation. Finally, both
MRK 1216 and NGC 1277 are anisotropic, with a global anisotropy parameter delta
of 0.33 and 0.58, respectively. While MRK 1216 follows the trend of
fast-rotating, oblate galaxies with a flattened velocity dispersion tensor in
the meridional plane of the order of beta_z = delta, NGC 1277 is highly
tangentially anisotropic and seems to belong kinematically to a distinct class
of objects.
