Whuttttt are you talking about???wth is a “private lawsuit”?? That’s not even proper legal terminology. An amicus means “friend of the court” - a person with an interest in the legal action but not an official party in the lawsuit can be a participant by filing a brief to have their interests represented in order provide their insight or expert opinion on the matter.
In doing so, they file an amicus brief whether it is the govt, corporation, private interest organization, nonprofit, citizen, public policy organization etc. Like
@HarlemHottie said, any person or group can file an amicus brief to petition the court. That is the literal purpose of the filing.