In 1944 German-occupied Poland, the child Erik Lehnsherr is separated from his parents upon entering a concentration camp. While trying to reach them, he causes a set of metal gates to bend towards him, as though attracted by a magnetic force, before being knocked out by guards. Decades later, U.S. Senator Robert Kelly attempts to pass a "Mutant Registration Act" in Congress, which would force mutants to publicly reveal their identities and abilities. Present are Lehnsherr, now known as Magneto, and the telepathic Professor Charles Xavier, who privately discuss their differing views on the relationship between humans and mutants.
In Meridian, Mississippi, 17-year-old Marie D'Ancanto accidentally puts her boyfriend into a coma upon kissing him, which is caused by her superhuman ability to absorb the life force and mutant abilities of anyone she touches. In fear, Marie, now going by the name Rogue, runs away to Laughlin City, Alberta. While at a bar, she meets Logan, also known as Wolverine, who possesses superhuman healing abilities, heightened senses, and metal claws that protrude from his knuckles. While on the road together, they are attacked by Victor Creed / Sabretooth, another mutant and an associate of Magneto. Cyclops and Storm arrive and save Wolverine and Rogue and bring them to the X-Mansion in Westchester County, New York. They are introduced to Xavier, who leads a group of mutants called the X-Men, who are trying to educate young mutants on their powers, and stop Magneto from escalating the war with humanity.
Senator Kelly is abducted by Magneto's allies Toad and the shapeshifter Mystique and brought to their lair, where Magneto uses Kelly as a test subject for a machine that artificially induces mutation. Kelly uses his new mutant abilities to escape imprisonment. After Rogue uses her powers on Wolverine, she is convinced by Mystique (disguised as classmate Bobby Drake) that Xavier is angry with her and she should leave the school. Xavier uses his mutant-locating machine Cerebro to find Rogue at a train station. Mystique later infiltrates Cerebro and sabotages the machine.
At the train station, Wolverine convinces Rogue to stay with Xavier, but a fight ensues when Magneto, Toad and Sabretooth arrive and kidnap Rogue. Kelly arrives at Xavier's school, but dies shortly after due to the instability of his artificial mutation, which causes his cells to break down into a puddle of water. The X-Men learn that Magneto was severely weakened while testing the machine on Kelly, and realize that he intends to use Rogue's power-transferring ability so that she can power the machine in his place, which will kill her. Xavier attempts to use Cerebro to locate Rogue, but Mystique's sabotage causes him to fall into a coma. Fellow telekinetic/telepath Jean Grey fixes Cerebro and uses it, learning that Magneto plans to place his mutation-inducing machine on Liberty Island and use it to mutate the world leaders meeting for a summit on nearby Ellis Island.
The X-Men scale the Statue of Liberty. Storm electrocutes Toad, and Wolverine stabs Mystique. Magneto transfers his powers to Rogue, and forces her to use them to start the machine. Cyclops dispatches Sabretooth with the help of Jean, who levitates his battle visor. Storm uses her weather-controlling powers and Jean uses her telekinesis to lift Wolverine up to Magneto's machine. Wolverine saves Rogue when Cyclops blasts Magneto out of the way, and destroys the machine. Wolverine touches the dying Rogue's face, and his healing abilities are transferred to her, causing her to recover.
Professor Xavier recovers from his coma. The group learns that Mystique is still alive, and impersonating Senator Kelly. Xavier tells Wolverine that near where he was found in Canada is an abandoned military base at Alkali Lake that might contain information about his past. Xavier visits Magneto in a prison cell constructed entirely of plastic, and the two play chess. Magneto warns him that he will continue his fight, to which Xavier promises that he and the X-Men will always be there to stop him.

Question: After the big fight what do the X-Men discover of Mystique?
Answer: That she is still alive and impersonating Senator Kelly.
Alternative answer: The liberty island fight