A union of states with a common recognised authority is
Answer Details
The term you are looking for is a federal system. In a federal system, a group of states, regions, or provinces come together to form a union under a common, central government that has authority over the entire nation. However, each state maintains its own government and has certain powers that are not overridden by the central authority. This system allows for a balance of power between the central government and the individual states. Examples of countries with a federal system include the United States, Canada, and Germany.
In contrast, a unitary system is one where a single central government holds most of the power, and any administrative divisions benefit from devolved powers that can be revoked by the central authority.
A representative system is a form of government where citizens elect representatives to make decisions on their behalf, but it does not inherently describe the relationship between a central authority and regions or states.
A confederal system is a loose alliance of independent states where the central authority has limited power, and the individual states retain the majority of governmental power. It is essentially the opposite of a unitary system.