A one-joke movie can be fun if the joke is a good one decently delivered, and that’s the case with “Elvis & Nixon,” a cheerily absurd take on the notorious meeting between the King of Rock and the President of the United States on December 21, 1970, which resulted in a still that the National Archives reports is among its most requested photos. The script by Joey Sagal, Hanala Sagal and Cary Elwes uses the details of the session reported by some of the participants as a skeleton for a largely imaginary satirical look at the relationship between politics and celebrity that’s grown ever stronger over the years since. The result may be more a sketch than a fully fleshed-out narrative, but it’s an engaging one, especially for those old enough to remember the actual event.

Much of the enjoyment comes from watching two of the best actors working today sparring with one another. Michael Shannon doesn’t much resemble Elvis Presley, nor does Kevin Spacey Richard Nixon, but they both have fun with the roles. Shannon depends more on a brusquely deadpan delivery (as well as the extravagant costumes) for the effect, while Spacey uses body language—the hunched shoulders, the beetle-browed way of staring people down, the awkward movements, to mimic the man’s outward look. Depth of characterization isn’t applicable in either case; both men are satisfied with simplicities, with Presley presented as a man blissfully aware of his superstardom and willing to make use of it with easygoing charm but without bluster, and Nixon as an insecure square who’d clawed his way to the top from humble beginnings and now jealously guards himself against perceived slights and suspected enemies. This is no profound psychological profile.

And the story itself is a slender one. As presented here, with a great deal of comic speculation, Elvis, who enjoyed carrying firearms and had a collection of badges from local law agencies, came up with the idea of getting a federal one—and, if he was really serious, using it to act as an undercover drug-enforcement agent—while watching news broadcasts about anti-war protests. So he flew on his own to Los Angeles, where he enlisted former aide Jerry Schilling (Alex Pettyfer) in a plan to fly to D.C. and meet with Nixon (and perhaps J. Edgar Hoover) to pitch his plan. Schilling, trying to make it on his own in the movie business and at the point of meeting his girlfriend’s parents, was reluctant to accompany Presley on what seemed a quixotic quest, but agreed out of a sense of loyalty. Meanwhile Elvis invited another old friend, spacey Sonny (Johnny Knoxville) to join them.

Reporting the appearance of Presley at a White House gate, where he asked that his handwritten request for a meeting be delivered to the President, to Nixon’s chief of staff Bob Haldeman (Tate Donovan) were his underlings Bud Krogh (Colin Hanks) and Dwight Chapin (Evan Peters), whose enthusiastic response to the proposal was initially dismissed by Haldeman, and then by Nixon. It was the support of the President’s daughter Julie, a big Presley fan who desperately wanted the King’s autograph, that sealed the deal according to this account. While the negotiations were continuing, however—a process that included a meeting between Presley’s men and the President’s in a parking garage (a sequence obviously tweaking the Deep Throat mythos)—Elvis also ambled into the D.C. office of the Bureau of Narcotics and Dangerous Drugs to ask a flustered official (Tracy Letts) for his help in smoothing the way.

Eventually the meeting is scheduled, and Presley and his little entourage show up—much to the consternation of the Secret Service—not only bearing arms themselves but carrying a commemorative pistol as a gift for Nixon. The President, still grumbling, intends only a brief session, and Presley intends no photos, but though Elvis breaks every rule of protocol Krogh had carefully stipulated—don’t sit on the sofa, don’t touch the M&Ms or the Dr. Pepper—the two hit it off and the meeting drags on, with Jerry and Sonny eventually joining in, and Presley ultimately gets everything he wants from the President, whose desperation to seem cool—especially in the eyes of his daughter—becomes ludicrously evident.

This is pretty thin stuff, to be sure, but it’s kept to a minimal running-time (86 minutes), and Spacey and Shannon have such a good time, particularly in their scenes with one another, that their enjoyment is contagious. Liza Johnson’s direction is more functional than imaginative, but she gives them free rein. Of the rest of the cast, Pettyfer’s restraint works best, though the subplot about his anxiousness to get back to L.A. gets a bit old even if it leads up to a good final joke. Hanks also scores with his mixture of boyish earnestness and harried subservience, as does Peters, though on a lower burn, and Letts gets in some good licks as the nonplussed BNDD official. Knoxville, as usual, comes across as more eagerly amateurish than professional. The tech credits are adequate down the line but not outstanding, though Terry Stacey’s camerawork proficiently captures the leads at work.

“Elvis & Nixon” is slight, but it moves briskly and gives Spacey and Shannon the chance to toy playfully with one of the nuttier episodes of American history.