French officials have called for vigilance but warned against jumping to conclusions after two weekend attacks - one in which a driver ran down 13 bystanders, and another in which a recent convert to Islam knifed police officers.
The driver, who had a history of psychological problems, was arrested following the attack in Dijon in eastern France, while the assailant who stabbed the officers outside the city of Tours was shot dead by police. None of the victims died.
The prosecutor in Dijon said the driver behind the attack in that city has a long history of severe mental illness and no links to terrorism.
The man, who is 40, has admitted his role in the attack, said prosecutor Marie-Christine Tarrare. She said he had been hospitalised 157 times for psychiatric problems since 2001, most recently this autumn.
She said the man, the French-born son of North African immigrants, acted alone and had no religious motivations, but rather was upset at the treatment of Chechen children.
He shouted "God is great" to give himself courage to act, and not out of religious belief, Ms Tarrare said.
The suspect was arrested. Eight people remain in hospital.
Interior minister Bernard Cazeneuve visited Dijon, an indication of the government's heightened concerns days after Islamic extremists renewed calls for individuals to strike in the West.
Counter-terrorist police are investigating the attack on police in a suburb of Tours on Saturday, which left two officers seriously injured and a third with light injuries.
"The president calls for solidarity with the victims, the families, both for the pedestrians in Dijon and the officers in Joue-les-Tours," said Stephane Le Foll, the government spokesman. "He also emphasised that we should not give in to panic."
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