Houston police have identified the suspect in the slaying of Dr. Mark Hausknecht as a man with a 20-year grudge against the physician, Houston Police Chief Art Acevedo said.
Police say Pappas' mother died on Hausknecht's operating table during surgery.
Dr. Mark Hausknecht, 65, was shot dead as he rode his bike in Texas on July 20. Authorities said the shooter rode past Hausknecht before turning around and firing.
Houston Police Chief Art Acevedo said that authorities have an arrest warrant for Pappas, who is armed, unsafe and potentially suicidal, NBC reported.
Hausknecht treated Bush in February 2000 for an irregular heartbeat after the ex-president complained about lightheadedness while visiting Florida.
Police posted images on Twitter in which Hausknecht is highlighted with a green circle and the suspect with a red circle.
A suspect sketch that police later released depicted a man wearing glasses and a cap. The Houston Police Department said the house was cleared just after midnight Friday, and that their search for the ex-deputy constable continues.
CNN called a phone number listed for Pappas' real estate company Wednesday, but the call went to voice mail.
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Pappas is also believed to be suicidal, in part because he sent a text message to an unidentified person saying he was going to kill himself, Acevedo said, adding that no one had heard from the suspect in days.
The Harris County District's Attorney's Office issued a murder warrant for Pappas' arrest Wednesday.
That phone number is linked to recent listings on a firearms auction website for several guns, ammunition, tactical vests and ballistic plates for auto doors.
Pappas took law enforcement classes through August previous year, documents indicate.
"The man who healed so many hearts during his life, has left so many broken by his death", a Houston Chronicle obituary read.
Investigative reporter Jeremy Rogalski uncovered the online ads on national firearms marketplace sites and the Houston Greensheet.
The ads on Craigslist.com, Armslist.com and this week's Houston Greensheet include listings on July 22 for two bullet-proof vests, and listings last week for a $4,500 semi-automatic rifle, a pistol and a bullet-proof panel for a Crown Victoria.
Hausknecht was a cardiologist at the Texas Medical Center. "People should not try to deal with him directly". But police believe 20 years of resentment harbored by the son of one of his former patients boiled over and led to the killing.