Standing Senate Rules XXI, XXIX, and XXXI cover secret sessions for legislative and executive business (nominations and treaties).
Rule XXIX calls for Senate consideration of treaties to be conducted in secret unless a majority votes to lift the “injunction of secrecy,” which it usually does.
Rule XXXI mandates that all nominations be considered in open session unless the Senate votes to do so in secret.
Rule XXI calls for the Senate to close its doors once a motion is made and seconded.
All of a chamber’s normal rules of debate apply during secret sessions, except during impeachment deliberations in the Senate.