A Cessna 560 responded to an apparently phantom TCAS RA shortly after takeoff.

Narrative

Shortly after takeoff; as we were climbing through 2000ft MSL; a TCAS RA alerted us to monitor vertical speed with a red indication on the positive climb side of the VSI; immediately followed by a descend RA. At the same time we were given a turn on course and a climb to a higher altitude which we immediately declined and we informed ATC that we were unable because we were responding to an RA. As we were descending back down; the 'don't sink' terrain warning was triggered; and we leveled-off around 2000 ft. We reset the TCAS/Transponder by going to standby and back to TA/RA which eliminated all of the advisories and warnings. Once we confirmed visually and with the controller that there was no traffic around us; we then accepted a higher altitude and continued without further complications. After talking to the controller; we were 100% positive that there was no traffic anywhere close to us and that it was simply a glitch in the equipment which was corrected by the recycle of the TCAS.