The incident occurred about 7:30 p.m. Thursday near the intersection of Whittier Boulevard and Indiana Street.
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"The force of the collision caused the deputy's patrol vehicle to run off the roadway and hit three pedestrians, a mother and her two children," the Los Angeles Police Department, which is conducting an investigation, said in a statement.
Surveillance video shows end of fatal Boyle Heights crash
According to the sheriff's department, deputies were responding to a gunshot call at the time of the crash.
The LAPD said the fatally struck boys were 7 and 9. One died at the scene; the other was pronounced dead at a hospital, where their mother was listed in critical condition. Family members identified the boys as Marcos and Jose Luis Hernandez.
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"The second car struck a third vehicle, causing injury to two additional adult pedestrians in the crosswalk," the LAPD said.
A total of seven people were transported to hospitals. Among them were five who suffered minor injuries -- two deputies and three civilians.
Family members gathered at the site of the crash, where a makeshift memorial with candles was set up.
"I don't have no words right now. You should understand what we're going through right now. It's painful," said Luis Hernandez Jr., the brother of the critically injured woman.
Anyone with information about the collision is asked to call the LAPD's Multidisciplinary Collision Investigation Team at (213) 833-3746.