We present a multiwavelength study of the AGN-host starburst galaxy PKS 0529-549 at z~2.6. We use (1) new ALMA observations of the dust continuum and of the [CI] 370 um line, tracing molecular gas, (2) SINFONI spectroscopy of the [OIII] 5007 Ang line, tracing ionized gas, and (3) ATCA radio continuum images, tracing synchrotron emission. Both [CI] and [OIII] show regular velocity gradients, but their systemic velocities and position angles differ by ~300 km/s and ~30 degrees, respectively. The [CI] is consistent with a rotating disc, aligned with the dust and stellar continuum, while the [OIII] likely traces an outflow, aligned with two AGN-driven radio lobes. We model the [CI] cube using 3D disc models, which give best-fit rotation velocities V~310 km/s and velocity dispersions sigma<30 km/s. Hence, the [CI] disc has V/sigma>10 and is not particularly turbulent, similar to local galaxy discs. The dynamical mass (~10^11 Msun) is comparable to the baryonic mass within the errors. This suggests that baryons dominate the inner galaxy dynamics, similar to massive galaxies at z=0. Remarkably, PKS 0529-549 lies on the local baryonic Tully-Fisher relation, indicating that at least some massive galaxies are already in place and kinematically relaxed at z~2.6. This work highlights the potential of the [CI] line to trace galaxy dynamics at high z, as well as the importance of multiwavelength data to interpret gas kinematics.
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