There's a scene early in the original X-Men film (2000) where Logan (Hugh Jackman) meets Professor X (Patrick Stewart) for the first time, and jokingly asks, "What do they call you, 'Wheels?'" 14 years later, Wolverine travels to the past to meet a younger professor (played by James McAvoy) and should be thinking the same thing - except something's off - somehow, Xavier can walk just fine despite taking a bullet to the back in X-Men: First Class.
In the climactic battle at the conclusion of Matthew Vaughn's X-Men prequel, Magneto deflects a bullet that strikes young Charles, and we later see him confined to a wheelchair. One of the world's most powerful mutants could no longer walk. In the marketing for the followup, X-Men: Days of Future Past, we see the same Xavier, years later, in a chair but also on foot once again. For those parts of the movie, what happened?
Another new clip for X-Men: Days of Future Past released by Twentieth Century Fox in the final weeks leading up to the May 23rd release of their expensive superhero team-up showcases a conversation between Xavier and Erik Lehnsherr which reveals the answer - or at least confirms what some of the cast have hinted at in previous interviews. Xavier has some sort of medication which alters his DNA and seemingly repairs his spine. He can walk and run once again, but with medication effecting his DNA, he has lost the effects the X-Gene has on his body and mind, effectively making him more Homo sapien than Homo superior.
The need for Xavier to walk has to do with the story of involving him in much of the plot, and the logistics of that and wheelchair accessibility gave the writing team room to flesh out his character journey - having him learn and choose what's important for him, his people and the greater good. Eventually he must accept that his mutant gifts are more important than the use of his legs. Magneto does a good job of calling our Charles and naming a pile of First Class characters who may have not survived through the '60s and into the early '70s.
The explanation for how this Xavier walks however is odd given the plot of X-Men: The Last Stand searching for a "cure" to mutant abilities. It seems we have already found it in whatever McAvoy's young Charles Xavier is on. Someone call Anna Paquin's Rogue! We expect that the same serum (or whatever he's taking) might be the same drug Beast is taking to return to his human form - another continuity-cracking plot point. Why make Beast blue and Xavier paralyzed, if they're going to write around it in the sequel? The same can be said for older Wolverine having his adamantium claws returning in the future.
The other question of continuity has to do with the older Professor X in the future who is miraculously alive despite being vaporized by Phoenix in X-Men: The Last Stand. That, according to Stewart, will be explained in the film. Clearly this all has to do with time-travel and alternate timelines, and how it's used may answer all of our questions. It's the plot device that drives the story of Days of Future Past and the one we hope makes sense of the franchise timeline(s) we've seen so far.
Here's another X-Men: Days of Future Past clip of non-blue Beast meeting Logan if you've not seen it:
Bryan Singer directs X-Men: Days of Future Past with newcomers Peter Dinklage, Omar Sy, Boo Boo Stewart, Fan Bingbing, Evan Peters, Adan Canto, Josh Helman, Evan Jonigkeit along with returning stars Jennifer Lawrence, Michael Fassbender, James McAvoy, Nicholas Hoult, Patrick Stewart, Ian McKellen, Hugh Jackman, Anna Paquin, Ellen Page, Shawn Ashmore, Halle Berry, Lucas Till, and Daniel Cudmore.
X-Men: Days of Future Past arrives in U.S. theaters on May 23rd, 2014, followed by Fantastic Four on June 19th, 2015, X-Men: Apocalypse on May 27th, 2016, The Wolverine 3 (not the official title) on March 3rd, 2017, Fantastic Four 2 on July 14th, 2017, and an unspecified Marvel film on July 13th, 2018.
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