"Exclusive activity" is the most general way to translate it, but depending on the context different phrases would be more natural/appropriate.
"Filing petitions in this court is the exclusive right of Attorneys licensed by the State Bar."
"Prescriptive authority is the sole prerogative of doctors and, in some cases, nurse practitioners."
"The Senate shall have the sole Power to try all Impeachments." - US Constitution
"No one agency has sole mandate over environmental regulation in the United States."
But these would sound very formal. It is more common to say things like:
"Only doctors and nurse practitioners can issue prescriptions."
"Only licensed professional engineers are allowed to offer engineering services to the public."