In the film, Alita: Battle Angel, the United Republic of Mars, or URM, was the governing body of Mars and the other human colony offshoots that joined Mars as member states prior to The Fall. The URM began as an offshoot of the Imperium, or the Earth Government, having gained its own Independence from the Earth government and then claiming and settling colonies of its own.
In the early 23rd Century, a bitter rivalry formed between the URM and the Imperium which resulted in a Cold War. After escalation, the war resulted in a devastating interplanetary war that destroyed the interstellar colonies and resulted in the loss of almost all life on Mars. The United Republic of Mars and its governments are presumed destroyed following the Fall; although with the isolation on Earth following The Fall, no one is certain of the fate of everyone on Mars and the other URM territories.
In spite of the United Republic of Mars being destroyed and the human race being plunged into a Dark Age following The Fall. Many of the technologies developed by the URM survived, including an unknown craft (possibly a bomber or transport), which contained the berserker body later used by Alita. Always the Damascus Blade, which Alita acquired from Zapan.