It would be hard to imagine a company undertaking a prestigious foreign engagement in more testing circumstances than faced the State Ballet of Georgia in the big opening dance event of the Edinburgh International Festival. The audience at the Playhouse was rooting for them, not that any special indulgence was called for. They may have been dancing with heavy hearts, but you wouldn’t have known it.

Founded in 1935, the Georgian troupe has been directed since 2004 by the former Bolshoi prima ballerina Nina Ananiashvili (herself Georgian), who has set her dancers high standards and strengthened the repertoire. At 45, she led the opening performance of Giselle in what has long been one of her most celebrated roles. The quality of her interpretation has not dimmed. She is still among the best, with the refinement of her style, the breadth and detail of her dancing — none of the technical challenges any effort at all — and a compelling command of the character, from affecting, persuasive innocence to a tragic weariness of soul in her betrayal, then the spirituality of her forgiving ghost. She had a perfect partner in Vasil Akhmeteli, as Albrecht, his acting clear, his dancing manly, elegant and strong. They achieved a most moving performance.

The matinée cast, Nino Gogua’s Giselle and Lasha Khozashvili’s Albrecht, didn’t, for me, reach that peak of extra-special emotional rapport, yet their dancing couldn’t be faulted. As the second programme would show, they are both exceptional artists. The peasant pas de six has new choreography by Alexei Fadeyechev of the Bolshoi, delightfully danced — the brothers Lasha and David Khozashvili neatly paired in the first cast. Some of the first-act plotting is overfussy, the acting stagey; and Viacheslav Okunev’s designs (with dinky-twee cottages) are not so successful as those he made for the Mikhailovsky Ballet’s Giselle, recently in London. Still, the production shows the company’s dancing quality. Robert W Cole conducted an attractive and dramatic account from the Scottish Chamber Orchestra.

For the mixed programme, three of the four pieces were given to recorded music, because the troubles at home had prevented Georgian musicians from getting here. From her connections of international stardom, Ananiashvili has given the repertoire a big boost in scope (they even dance Frederick Ashton’s La Fille mal gardée in Tbilisi) — and, fittingly, this includes a good deal of Balanchine. He came of a Georgian family, and his brother, the composer Andria Balanchivadze, was a co-founder of this company.

We opened with Chaconne, Balanchine’s lovely dances for Gluck’s Orfeo ed Euridici: dances of sweet serenity, then formal display, to a grand climax (though I am at a loss as to why, in these designs, one soloist wears a skirt like half a cabbage). The handsome leading couple were Anna Muradeli and Akhmeteli — her extensions unfolding creamily in his smooth lifts.

For Duo Concertant (Balanchine, set to Stravinsky), with David Tong and Benjamin Nabarro stepping in at short notice as pianist and violinist, Gogua and Lasha Khozashvili were beautifully matched in one of the best performances I have seen of this inventive and intricate duet.

Then Edinburgh had the UK premieres of two ballets created in February by well-known Russian and international choreographers of the new generation. Bizet Variations, by Alexei Ratmansky (lately the reforming director of the Bolshoi), has a romantic flavour — the women in chiffony dresses and elbow-length gloves, men in billowing blouses — with rivalry in the relationships of three couples. The dances are fluent, by turn pretty and vigorous, and Ananiashvili is the central figure to whom three men pay court. She is subtle, glamorous and palpably desolate when it seems she has been deserted. But no, all ends happily.

In Sagalobeli, by Yuri Possokhov (now resident choreographer of the San Francisco Ballet), classical ballet is blended with steps from Georgian folk dance. The attractive music by the Georgian ensemble Changi — who were to have performed live, though instead we heard their recording — uses native traditional instruments, varieties of harp, lute, flute and drums, as well as polyphonic singing. The Khozashvili brothers, in jerkins and boots, open with a curious duet, a test of strength involving a rope; then women bourrée in a line, with dipping heads and graceful arms, a charming effect done in silhouette. In patterned dresses, they shimmy, ripple and glide. There are round-dances and flat-foot processions. The men erupt in a macho display of technical bravura that builds (to drums) as if they are possessed. This piece — the climax to a well-chosen programme — wowed the audience, from whom the Georgians will take home admiration and affection.