If the sauce won't be served for a while after it's finished, then you have to keep it warm, but it must be ≤ 88 °C / 190 °F. Put a tight-fitting lid on it, and put the stove burner at its lowest setting.
Don't add the butter too quickly, or it will drop the temperature, which will make it break. (unless you melt the butter ahead of time)
If you have some left over, you can put it in the fridge. However, cooling it does make it break, so it can only be treated as regular butter (if possibly flavored) after that.
Beurre rouge — Beurre blanc, but using red wine instead
Monter au beurre — It's a different way to emulsify butter in a sauce. "Monter au beurre is the process of adding cold butter to a sauce after you’ve taken it off the heat. Adding the chilled butter produces a similar emulsification effect, adding body and shine to an existing sauce."
? — beurre monte, but with a stock (vegetable, chicken, dashi, etc) instead of water