‘The Golden Apple,” a critical hit and commercial flop in 1954, is ending the New York City Center Encores! season with strenuous ambition and serious talent in a cornball musical-comedy hodgepodge.

This seems about right for a Greco-Americana oddity — and cult favorite — that bounces bits and characters from the Iliad and the Odyssey into a turn-of-the-last-century farm town full of sophisticated arcane winking and over-the-top homespun goofballs mugging through director Michael Berresse’s uneven staging and Joshua Bergasse’s mostly coarse choreography.

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Scenes leap from syncopated square-dance music to jazz to operetta. There are so many multiple personalities that one would think, if we didn’t know better, that its creators were afraid they’d never get a chance to write another musical. On the contrary, of course, the score is by Jerome Moross, protégé of Aaron Copland and movie composer of “The Big Country.” Book and lyrics are by John Latouche, librettist for provocative 20th century operas.

These days, however, the show — the first musical to transfer from off-Broadway to Broadway — is best known for “Lazy Afternoon.” The languorous, seductive ballad is sung here by Lindsay Mendez, amusing as Helen, the racy married farmer’s daughter who runs off to the big city with Paris, a salesman who arrives in a balloon and is embodied with riveting ballet form and charm by Barton Cowperthwaite.

Ulysses (strapping baritone Ryan Silverman) arrives from the Spanish-American War with the other “boys in blue” (rhymes with hullabaloo), who have such familiar names as Agamemnon and Achilles and who, for reasons unknown, briefly turn into a baseball team. Ulysses loves his loyal wife, Penelope — the classically trained Mikaela Bennett, a major young talent who maneuvers beautifully through the operetta and the rich Gershwin-inspired songs.

But, you know, the pleasures of the home can’t compete with heroic adventures, which take a wicked toll in his 10-year odyssey in the more interesting second act. There, the ever-concerned gypsy seer (the impressive single-named N’Kenge) transforms into Circe, and Ulysses, finally chastened, sings a soliloquy that inquires nothing less than “what is the meaning of life?” and “what is the mystery of death?”

This is a lot to ask from a musical that includes a pie-and-cake contest and actors directed to act out every word, especially ones about sex, in case we don’t get it. Once every 63 years, I guess that’s OK.