He has no hymn, no described act, no attribute and no epithet, and appears almost only inside enumerations of gods. Aja is either a goat or, read as a-ja, the unborn, and ekapada is one-footed or having one support. Readings were proposed in the nineteenth century and nobody has defended any of them at length since, because there is no passage anywhere against which one could be tested. This is a gap rather than a disagreement. He appears only in enumerations, almost always beside Ahi Budhnya, so no domain can be read off the surviving text either.
What would resolve this A passage from outside the Rigveda that describes rather than merely names him, whether in the Brahmana literature, the Atharvaveda or a comparative Indo-Iranian source; or a demonstration that the formulaic pairing with Ahi Budhnya is itself the whole content of the name, which would close the question by showing there is nothing behind it. A comparative Indo-European study connecting the name to a documented mythic role would serve equally.
OpenVedic philology and comparative Indo-Iranian religionArthur Anthony Macdonell, Vedic Mythology