Laurie Anderson, the American musician and artist, also threatened to pull out. “I have a big problem being part of a festival that asks artists to distance themselves from their beliefs and commitments — whatever they are,” Ms. Anderson said in an email.

Faced with the risk of losing a headline act like Ms. Anderson — and possibly others — the festival’s director, Stefanie Carp, did an about-face. On June 21, she announced that she had asked the Young Fathers to rejoin the lineup, “although I do not share their attitude to the B.D.S.”

“I believe that we need to allow the different perspectives and narratives,” Ms. Carp said in a statement posted on the festival’s website.

The band immediately rejected the offer, according to their manager, James Stanson, who said in a telephone interview that they felt uncomfortable returning to an event that had rejected them. In an earlier statement, the band said it was “wrong and deeply unfair” for a festival to “ask us to distance ourselves from our human rights principles in order for the appearance to go ahead.”

The band will not play any events in Germany that receive public funds for the foreseeable future, Mr. Stanson added, because of the risk of similar problems.

Indeed, conflict over B.D.S. has already begun to spread.

A festival in the northwestern city of Osnabrück rejected calls to bar a Syrian electronic music producer, Samer Eldahr, who performs as D.J. Hello Psychaleppo, by a protest group that said Mr. Eldahr supported B.D.S. In a statement released by the festival, Mr. Eldahr denied being a member of the movement. “To equate criticism of the policy of the state of Israel with anti-Semitism is simplistic and dangerous,” the festival director added.