In a grainy black-and-white photograph from 1968, Suzanne Farrell, gently supported by Arthur Mitchell, both of them then principals at the New York City Ballet, arches her back in a perfectly curved arabesque. Shot during the company’s halcyon days, it advertised Balanchine’s “Pithoprakta,” which he choreographed to a score by the Greek composer Iannis Xenakis.

This haunting image and a barely decipherable videotape were all that remained of the striking work until financial support from the Balanchine Preservation Initiative allowed Ms. Farrell to reconstruct it for her troupe, the Suzanne Farrell Ballet, which performed the ballet for the first time at the Kennedy Center in Washington, D.C., last year. New Yorkers will have an opportunity to see it on September 25 at the Fall for Dance Festival at City Center.

“This ballet was a departure point for Mr. B,” Ms. Farrell said recently by phone. “At the time, astronauts were about to go to the moon. He was fascinated by the possibilities of new worlds, and that fascination sparked the off-kilter movement and unusual writhings and twistings. It was very radical at the time. It looks like it could have been choreographed today.”

Ms. Farrell has devoted the past 23 years to preserving ballets that Balanchine created for her, some of which he willed to her at his death in 1983. Appointed repetiteur by the Balanchine Trust, an independent organization founded to oversee the licensing and staging of his ballets, she brought them to the Kirov Ballet, the Paris Opera Ballet, the Royal Danish Ballet and many American companies. His most important muse — he choreographed almost two dozen ballets for her, including “Don Quixote,” “Jewels,” “Vienna Waltzes,” and “Mozartiana” — she knows his works better than anyone. Luckily for audiences and posterity, she has been able to see that they are performed as he would have wanted, especially since establishing her company in 2000 at the Kennedy Center. Her troupe performs more than 35 ballets by Balanchine and her other mentors, Maurice Béjart and Jerome Robbins. Professor of Dance at Florida State University, she won a prestigious Kennedy Center Honor in 2000.

Established in 2007, the Balanchine Preservation Initiative has enormously helped Ms. Farrell with its mission to introduce rarely seen or lost Balanchine works to audiences around the world, many of which have not been performed in nearly 40 years. “Balanchine was our greatest choreographer,” the director of development at the Kennedy Center, Daniel J. Hagerty, said. “Losing these ballets would be tragic. Suzanne is making sure that we won’t.” Her company’s repertoire now includes nine Balanchine Preservation Initiative ballets such as “Variations for Orchestra,” “Divertimento Brillante,” and “Pithoprakta.”

It is a painstaking process, particularly in the case of “Pithoprakta” — the Greek title means “action by probabilities” — the quality of the videotape is so poor. It was originally danced as the second part of a ballet to the Xenakis score, “Metastasis & Pithoprakta.” Adding to the challenge, the principal male dancer, Mr. Mitchell, apparently wasn’t available when the tape was made, so Ms. Farrell couldn’t see what he danced. And neither she nor Mr. Mitchell, who is artistic director and co-founder of Dance Theatre of Harlem, remembered much of the choreography.

Undaunted, Ms. Farrell reconstructed the ballet in her usual fashion by going into the studio and performing what she could recall, all the while carefully watched by her dancers. Like Balanchine, she doesn’t explain very much when she puts together a piece, preferring to demonstrate what she wants. Xenakis’s score calls for more than 50 instruments each playing something different, from musical sounds to sound effects. “When I heard the music again,” she said, “some vibrations came back to me and I worked from those. Mr. B was very interested in the architectural and mathematical structure of the music. The movement reflects the score’s unpredictability. At times, it’s very spiky, angular, and chaotic.”

She knew that there was little interaction between the male and female leads. The toughest part was re-creating the male solos. “Whenever there was a phrase of music where nothing seemed to be happening,” she said, “I’d think, ‘That must be where Arthur dances.'”

Ms. Farrell gave her original role to dancer Elisabeth Holowchuk, who compared the process of reconstructing the ballet to “piecing together a puzzle.” They went by trial and error, experimenting with different combinations, inventing sequences when necessary. For this ballet, Ms. Farrell also changed and updated the costumes, which originally featured pink rhinestones and gold spangles. The corps of seven women and five men now wears black unitards, and the two leads appear in white. In a stroke of genius, she asked for a black-and-white backdrop that resembles a mathematical diagram filled with masses of short black bars and numbers, a mural-size replica of the original Xenakis score, bars 52-560 for pizzicato glissandi. It gives the ballet a strong visual frame.

Ms. Farrell’s dancers study her every move and artistic decision. “Suzanne is as close as we can get to Balanchine,” Ms. Holowchuk said. “We feel him through her. What a gift.”