The French far-right leader, Marine Le Pen, has said she won't join other political parties in a march against anti-Semitism on Tuesday, accusing France's leaders of doing nothing to tackle Islamist networks in France and saying she will mark the occasion separately.
It comes days after a prominent French philosopher, Alain Finkielkraut, was verbally attacked for being Jewish as he walked past the weekly "gilets jaunes" (yellow-vest) protests in Paris.
A small group of protesters shouted a barrage of abuse at him as he passed by the demonstration on his way home from lunch on Saturday, calling him a "dirty Zionist" and telling him to "go back to Tel Aviv".
"I felt an absolute hatred," Mr Finkielkraut told one French newspaper later that night. "If the police hadn't been there, I would have been frightened."
A few days before that, official data suggested there had been a 74% rise in anti-Semitic attacks in France last year.
Now, many here are questioning whether the gilets jaunes movement is providing a new kind of forum for these extremist views, and how central those attitudes are to the movement.
"It's very serious," says Vincent Duclert, a specialist in anti-Semitism in France at the School for Advanced Studies in the Social Sciences - one of France's most prestigious colleges.
"The gilets jaunes are not an anti-Semitic movement, but alongside the demonstration each Saturday there's a lot of anti-Semitic expression by groups of the extreme right or extreme left."
"You can be on the streets demonstrating every Saturday, shouting your slogans against the Jews," says Jean-Yves Camus, an expert in French political extremism.
"And as there's no leadership in the movement and no stewarding of the demonstrations, you can be free to do it. I'm afraid there will be more attacks, because the self-proclaimed leaders simply do not seem to care that much."
Jason Herbert, a spokesman for the movement, says the incident on Saturday is a scandal, but not representative of the gilets jaunes as a whole.
"It's the inherent weakness of a movement that lets the people speak," he explained. "Everyone can come and give his opinion - and some opinions are despicable and illegal. To think someone is inferior because of his or her origins is just not acceptable, and it's completely unrelated to our demands. Amongst our demands, I've never heard 'we want fewer Jews'."
The gilets jaunes began life as a protest against fuel tax rises, but have broadened into a loose confederation of different interest groups with no official hierarchy or leadership. Over the past three months, as the movement has appeared more radical, its wider support has dipped.
Vincent Duclert believes that the movement does bear some responsibility for the extremist abuse in its midst, because the violence of the protests - towards the police, state institutions and public property - encourages anti-Semitism by encouraging "transgression".
And, he says, it's possible that the gilets jaunes are also offering "a new space for different kinds of anti-Semitism to come together: from the extreme right and extreme left, but also from radical Islamist or anti-Zionist groups, and some types of social conservatives".
There are signs over the past year, he says, that levels of anti-Semitism have risen within these different groups, because of changes at home, across Europe and in the Middle East, and that French public opinion has been too tolerant.
Politicians here have been quick to condemn Saturday's attack on Alain Finkielkraut. President Macron tweeted that it was "the absolute negation of what we are and what makes us a great nation".
Others tried to blame it on their political rivals.
A member of France's centre-right opposition, Geoffrey Didier, told reporters that anti-Semitism was growing "because radical Islamism is growing in France", while Marine Le Pen said it illustrated "how the anti-Semite far-left is trying to infiltrate the gilets jaunes movement".
Both Ms Le Pen's party and that of her far-left rival, Jean-Luc Mélenchon, have been trying to win the support of the gilets jaunes ahead of European elections in May.
Jean-Yves Camus believes last week's attack will help turn public opinion against the movement, saying it has become "a hotbed of radical activity from both sides of the political spectrum and the French do not want that".