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A gunman who killed eight people at a Texas shopping mall last weekend appeared to have neo-Nazi beliefs, but targeted people at random, law enforcement officials have said.

Police said the 33-year-old suspect appeared to have chosen his victims with no regard for their race or sex.

He had not been in trouble with the law before Saturday's attack in the city of Allen, police said.

Investigators are reviewing his social media.

The suspect, Dallas resident Mauricio Garcia, was shot dead at the scene by a police officer.

"We do know he had neo-Nazi ideation," Texas Department of Public Safety (DPS) Regional Director Hank Sibley told a news conference on Tuesday. "He had patches. He had tattoos."

Visible during the attack was a clothing patch the killer wore with the letters RWDS, which stands for "right-wing death squad", a phrase popular among white supremacist groups.

"To me, it looks like he targeted the location rather than a specific group of people. He was very random in the people he killed, it didn't matter the age, race or sex," said Mr Sibley.

"The big question that we're dealing with right now is what's his motive, why did he do this? Well, the big question is, we don't know."

He added that the man had legally bought eight weapons, including the AR-15 he was seen firing. Three firearms were found in his possession, while another five were in his vehicle.

The Anti-Defamation League (ADL) Center on Extremism said on Tuesday that research into the suspect's social media indicate he was "obsessed with violence and subscribed to a range of extremist ideologies, including antisemitism, violent misogyny and white supremacy".

Mr Sibley also confirmed reports of the gunman's brief stint - and discharge - from the US Army.

"Early on in basic training, they realised, from what I've seen, that he was not a fit for the army," he said, adding "there were some questions about his fitness for duty".

According to the US Defence Department, the suspect entered the army in June 2008 and was "terminated three months later without completing initial entry training" because of "physical or mental conditions".

His weekend attack, which took the lives of three children as young as three and left seven other people injured, ended when a police officer responding to an unrelated call killed him on the scene.

The suspect had by then shot 15 people inside the crowded Allen Premium Outlet Malls within the space of three or four minutes, according to the Texas DPS.

Zach Horn, an attorney for the responding officer, released a statement to local media lauding him as "a brave servant" who had "sprinted towards high power rifle fire as everyone else ran away".

"He's doing well and would appreciate privacy as he continues to process this life-altering tragedy," Mr Horn said.