Transition disks are protoplanetary disks with inner cavities possibly cleared by massive companions. Observing them at high resolution is ideal for mapping their velocity structure and probing companion-disk interactions. We present Atacama Large Millimeter/submillimeter Array (ALMA) Band 6 dust and gas observations of the transition disk around RXJ1604.3-2130 A, known to feature nearly symmetric shadows in scattered light, and aim to search for non-Keplerian features. We studied the 12CO line channel maps and moment maps of the line-of-sight velocity and peak intensity. We fitted a Keplerian model of the channel-by-channel emission to study line profile differences and produced deprojected radial profiles for all velocity components. The ^12^CO emission is detected out to R~1.8" (265au). It shows a cavity inward of 0.39" (~56au) and within the dust continuum ring (at ~0.56", i.e., 81au). Azimuthal brightness variations in the ^12^CO line and dust continuum are broadly aligned with the shadows detected in scattered-light observations. We find a strong localized non-Keplerian feature toward the west within the continuum ring at R=41+/-10au and PA=280+/-2deg. It accounts Dv_phi_/v_kep_~0.4 or Dv_z_/v_kep_~0.04, depending on if the perturbation is in the rotational or vertical direction. A tightly wound spiral is also detected and extends over 300deg in azimuth, possibly connected to the localized non-Keplerian feature. Finally, a bending of the iso-velocity contours within the gas cavity indicates a highly perturbed inner region, possibly related to the presence of a misaligned inner disk. While broadly aligned with the scattered-light shadows, the localized non-Keplerian feature cannot be solely due to changes in temperature. Instead, we interpret the kinematical feature as tracing a massive companion located at the edge of the dust continuum ring. We speculate that the spiral is caused by buoyancy resonances driven by planet-disk interactions. However, this potential planet at ~41au cannot explain the gas-depleted cavity, the low accretion rate, and the misaligned inner disk, which suggests the presence of another companion closer in.