We report on a deep radio search for a pulsar wind nebula associated with the radio-quiet neutron star RX J0822-4300 in the supernova remnant Puppis A. The well-determined properties of Puppis A allow us to constrain the size of any nebula to less than 30 arcsec; however we find no evidence for such a source on any spatial scale up to 30 arcmin. These non-detections result in an upper limit on the radio luminosity of any pulsar-powered nebula which is three orders of magnitude below what would be expected if RX J0822-4300 was an energetic young radio pulsar beaming away from us, and cast doubt on a recent claim of X-ray pulsations from this source. The lack of a radio nebula leads us to conclude that RX J0822-4300 has properties very different from most young radio pulsars, and that it represents a distinct population which may be as numerous, or even more so, than radio pulsars.
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