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You have a couple of ways to do business here. First, you can form a new LLC, depending on the state you are in it could be a series LLC, which is one that is formed under your existing LLC, with your existing LLC as owner. Another way to do this is a Doing Business As (DBA
) or Fictitious Business Entity, but you have to register that with the secretary of state as well so ownership would be traced back to your original LLC.
For pure anonymity, the best way to do this is to form yourself a trust. The trust is not recorded anywhere as who owns it. The trust then can form yourself a new LLC with the trust being the owner and then nobody would ever know who owns the LLC other than the trust, but nobody would know who owns the trust.
You can also file an LLC in Nevada or WY, which are two states that allow only the former of the LLC name to appear and keeps the owner names confidential.
These are your legal options.