Is there a method to combine encryption and ring signing? There's the well-known question on whether to sign or encrypt first. However, if instead a message is to be ring-signed such that either recipient could be the actual signer, is there any scheme that combines signing and encrypting in one "atomic" step?
Yes, like there are conventional signcryption schemes combining digital signatures with encryption into a single primitive, there are also ring signcryption schemes that realize the same for ring signatures and encryption. Look for instance here or just ask Google scholar.