According to Rabbi Zilai, if one does not have oil with which to cleanse his hands after eating, this prevents him from reciting the Grace after Meals blessing.
Rabbi Zivai says: Lack of that oil does not prevent one from reciting Grace after Meals.
R Zuhamai says: Just as one who is filthy is unfit for Temple service, so too are filthy hands unfit for reciting the Grace after Meals blessing.
In response to this baraita R Naḥman bar Yitzḥak says:
I do not know of Zilai or Zivai or Zuhamai; rather, I know a baraita, as R Yehuda said that Rav said, and some say that it was taught in a baraita: It is stated: “And you shall sanctify yourselves, and you shall be holy, for holy am I, the Lord your God” (Vayikra 20:26). With regard to this verse, the Sages said: And you shall sanctify yourselves, these are the first waters with which one washes his hands before the meal; and you shall be holy, these are the final waters; for holy, this is oil which one spreads on his hands; am I, the Lord your God, this is the Grace after Meals blessing.
The Maharatz Chajes suggest that the strange names of these Sages, along with the fact that they do not appear in any other sources, leads to the conjecture that Zivai, Zilai, and Zuhamai are nicknames rather than the actual names of these individuals. It seems that certain Sages who only stated a single well-known halakha came to be known by that halakha. The halakha concerning disqualification due to filth, zohama, led to its author being called Rav Zuhamai. Apparently, that is the case with regard to Zilai and Zivai as well.