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We have examined several archival ASCA observations of Galactic radio pulsars, including PSRs B1046–58 and B1610–50. X-ray emission is detected from PSR B1046–58 with a significance of ∼5σ. We find no evidence for pulsations. We argue that the emission is from a spatially unresolved synchrotron nebula powered by the pulsar. The location of the X-ray counterpart within the 95% position error ellipse of the gamma-ray source 3EG J1048–5840 strengthens the claim of Kaspi et al. (2000) that PSR B1046–58 emit ϒ-rays. X-ray emission from PSR B1610–50 is not detected. We use the X-ray luminosity upper limit to constrain the pulsar’s velocity ≲ 200km s−1, arguing against an association with the nearby supernova remnant Kes 32. Our results for these radio pulsars contradict previous reports of detections of large (tens of arc minutes) associated synchrotron nebulae.