Tens of thousands of Danes gathered at torch-lit memorials around the country on Monday, commemorating victims of deadly attacks on a synagogue and an event promoting free speech that shocked a nation proud of its record of safety and openness.

Singing John Lennon's Imagine, defiant Danes promised to uphold their  open society and showed solidarity with the country's Muslim minority after reports the gunman was a Dane with Palestinian roots and a passion for Islamist issues.

The 22-year-old gunman opened fire on a cafe in hosting a free speech debate on Saturday, killing one, and attacked a synagogue, killing a guard. He was later killed in a shootout with police in his neighbourhood of Norrebro, a largely immigrant part of the city with a reputation for gang violence.

Police, which have not publicly the identified the gunman, arrested two people on suspicion of aiding the attacks but said there was no indication the shooter was part of a cell or had travelled to Syria or Iraq.

"We have now experienced the fear that terrorism seeks to spread," Prime Minister Helle Thorning-Schmidt told reporters. "The Danish democracy is strong, the Danish nation is strong, and we will not accept any attempt to threaten or intimidate our liberties and our rights."

Jewish leaders also called for calm and tolerance as some Muslims feared a backlash.

"We fight together with them (Muslims) for religious rights. We are moderates. We fight together against extremism and radicalism," Dan Rosenberg Asmussen, chairman of the Danish Jewish Community, told a press conference.

Thousands of Danes left flowers at the synagogue, walking in a quiet, solemn procession, with many also leaving both Danish and Israeli flags. A march by PEGIDA, the anti-Islam movement born in Germany, however, attracted only around 50 people.

Saturday's cafe event was attended by Swedish artist Lars Vilks, who has received death threats for drawings of the Prophet Mohammad, and by French ambassador Francois Zimeray, who likened the attacks to the Charlie Hebdo attacks in Paris. Vilks and Zimeray were both unharmed.

The killings shocked Danes who pride themselves on a welcoming and safe society, and fed into a national debate about the role of immigrants, especially Muslims. The populist Danish People's Party, which campaigned against the building of a mosque here, has strong support in the polls.

Denmark became a target of Islamists 10 years ago after the publication of cartoons lampooning the Prophet Mohammad, images that led to sometimes violent protests in the Muslim world.

FIGHTER, GOOD STUDENT

The gunman, named by Danish media as Omar Abdel Hamid El-Hussein, was well known to police for violence, weapons violations and his membership in a gang. Reuters could not confirm his identity and police declined to comment.

Police records show a man named El-Hussein was convicted of stabbing a man in the leg on a Copenhagen train in 2013 and Danish media said he was released from prison in January.

"He was 'normal' religious, nothing unusual, he didn't go to mosque any more than the average Muslim," El-Hussein's father told TV2.

He was an avid kick boxer in his younger years and was often known by the nickname "Captain Hussein" but members of his club said he has not been there for years.

"He was a good student," Peter Zinckernagel, El-Hussein's principal at the VUC Hvidovre school near Copenhagen told Reuters. El-Hussein attended the school until the end of 2013, when he was arrested for the train stabbing.

National news broadcaster TV2 said El-Hussein's parents were Palestinian refugees who came to Denmark after living in a Jordanian refugee camp for several years.

TV2 obtained a psychiatric assessment of El-Hussein conducted in connection with the assault case for which he was imprisoned in which he told psychologists he had a happy childhood and good relations with his parents and a younger brother. However, he did not graduate from school, was unable to get into a university and later was homeless.

Citing two unnamed friends, Politiken daily newspaper said the man was passionate in discussions about the Israeli-Palestinian conflict and had a short fuse. They expressed shock that he should launch such attacks, however.

Norrebro, where the gunman lived, mixes housing estates and seedy bars with bicycle paths and gardens. However, as house prices in Copenhagen risen, many young professionals have also made the area their home and it now hosts trendy music venues and at least one Michelin-starred restaurant.

The attacks raised questions about whether Denmark should tighten security measures. Even before the attacks, parliament debated whether to pass a law that would allow authorities to confiscate the passports of radicalised youths wanting to travel to the Middle East.