"So far there are no clues to a possible motive for the act," said Martin Botzenhardt, senior prosecutor in the northwestern city of Muenster where the attack happened, in a statement issued in the early hours.
"We are pressing hard on our investigation into all possible avenues."
Late Saturday, authorities were near-certain that there was no Islamist connection to the violence in the historic centre of Muenster as had initially been feared.
The two people 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 -- six of them seriously -- amid the broken and upturned tables and chairs seen strewn across the pavement in images of the scene.
Police had to wait for a bomb disposal team to clear the vehicle used in the attack after noticing suspect wires inside.
In the end, they found only the weapon used by the 48-year-old 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 about the attack, setting up a website where people can upload photos or videos.
- 'No Islamist connection' -
"There was a bang and then screaming. The police arrived and got everyone out of here," an employee of the restaurant hit by the terrace told NTV.
"There were a lot of people screaming. I'm angry -- it's cowardly to do something like this."
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.
Germany has been on especially high alert for jihadist attacks after several claimed by the Islamic State 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.
Media reports said the driver, identified only as Jens R., had a history of mental health problems.
Public broadcaster ZDF said the man 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 reported that he had possible links with far-right movements.
- 'Deeply shaken' -
Chancellor Angela Merkel said she was "deeply shaken" by the incident and "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, each 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 attack 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 in central Berlin.
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.//AFP