There are “no indications of a political motive,” said Hajo Kuhlisch, chief of police in western city Münster where the attack took place.
Rather, “the motive and origins (of the crime) lie within the perpetrator himself,” he added.
German investigators were still puzzling Sunday over the motives of the driver, who ploughed a van into a crowd at an open-air restaurant the day before, killing two people before shooting himself dead.
“So far there are no clues to a possible motive for the act,” said Martin Botzenhardt, Münster senior prosecutor, in a statement. “We are pressing hard on our investigation into all possible avenues.”
Authorities were near-certain that there was no Islamist connection to the violence in the historic centre of Muenster as had initially been feared. On Sunday they said they believed the driver had acted alone.
Media reports said the 48-year-old German driver, identified only as Jens R., had a history of mental health problems.
The two victims killed were a 51-year-old woman and a 65-year-old man, both from northern Germany.
As well as the dead, police said 20 were injured — some life-threateningly — amid the broken and upturned tables and chairs seen strewn across the pavement. The Dutch foreign ministry said two of those hurt were Dutch, one of whom was in a critical condition.
In the van, police found the gun used by the driver to kill himself, a blank-firing pistol and some powerful fireworks. A search of the man's Muenster apartment late Saturday turned up more fireworks and a deactivated AK47 assault rifle.
Police have appealed to the public for information, setting up a website where people can upload photos and videos.
'No Islamist connection'
Armed police cordoned off a wide area around the scene of the attack, urging residents to avoid the city centre to allow investigators to get to work amid initial fears the country had suffered another extremist assault.
“I was on my way home through the city here and saw firefighters and ambulances everywhere. I thought something really terrible must have happened,” said Hubert Reckermann, a local man in his late 60s. “It's still unbelievable for me, but these days anything can happen. You can't really defend yourself against people with psychiatric problems.”
Germany has been on especially high alert for jihadist attacks after several claimed by the Islamic State (Isis) group. But in the Saturday afternoon attack, inflicted as locals and tourists enjoyed a sunny spring day, there was “no indication at the moment that there is any Islamist connection,” said North Rhine-Westphalia state interior minister Herbert Reul.
Far-right politicians had sprung to denounce what they assumed was a jihadist attack, hoping to tap more political capital from Chancellor Angela Merkel's 2015 decision to open Germany's borders to more than one million mainly Muslim refugees and migrants.
While stressing the investigation was still ongoing, Reul said that the perpetrator was “not, as has been claimed everywhere, a refugee or something like that”.
Public broadcaster ZDF reported the driver had recently attempted suicide while rolling news channel NTV said he had spoken of a desire to bring as much attention as possible to his death.
ZDF also said that he had possible links with far-right movements.
Merkel said she was “deeply shaken” by the incident and pledged that “everything possible will be done to determine what was behind this act and to help the victims”.
The presidents of Russia and France, Vladimir Putin and Emmanuel Macron, as well as Spanish Prime Minister Mariano Rajoy sent their condolences.
The attack is the latest in a string across Europe in which vehicles have been used to attack crowds of people in public places. In a Berlin assault in December 2016, Tunisian asylum-seeker Anis Amri hijacked a truck and murdered its Polish driver before killing another 11 people and wounding dozens more by ploughing the heavy vehicle through a Christmas market. He was shot dead by Italian police in Milan four days later while on the run.
In France, the Islamic State group claimed a 2016 truck attack in Nice on its July 14 national holiday that killed 86.
And in Spain, the jihadists also claimed a rampage along Barcelona's Las Ramblas boulevard in August 2017 that killed 14 and left more than 100 injured.
By AFP's Tom Barfield