משנה אויף במדבר 19:3

One that is born from the side, the hire of a harlot or the price of a dog is invalid. Rabbi Eliezer says that it is valid, as it says, "You shall not bring the hire of a harlot or the price of a dog into the house of the Lord your God," (Deuteronomy 23:19) and this was not brought into the house. All blemishes that cause consecrated animals to be invalid cause also the [red] cow to be invalid. If one had ridden on it, leaned on it, hung on its tail, crossed a river by its help, doubled up its leading rope, or put one's cloak on it, it is invalid. But if one had only tied it up by its leading rope or made for it a sandal to prevent it from slipping or spread one's cloak on it because of flies, it is valid. This is the general rule: wherever anything is done for its own sake, it remains valid; but if for the sake of another, it becomes invalid.

If the cow refused to go out, they may not take out with it a black one lest people say, "They slaughtered a black cow" nor another red [cow] lest people say, "They slaughtered two." Rabbi Yose says: it was not for this reason but because it is said "And he shall bring her out" by herself. The elders of Israel used to go first by foot to the Mount of Olives, where there was a place of immersion. The priest that was to burn the cow was (deliberately) made unclean on account of the Sadducees so that they should not be able to say, "It can be done only by those on whom the sun has set."