Investigators from Egypt and Israel are conducting a joint on-site examination after a shooting that left three Israeli soldiers and an Egyptian police conscript dead.
Egypt said a member of its security forces “breached the border security barrier” while pursuing drug smugglers.
Authorities are conducting background checks on the police conscript and his family and friends to try to establish a motive for crossing the border, other than the pursuit of drug smugglers.
The officials, who spoke to The National anonymously, identified the conscript as Mohammed Salah Ibrahim, 23, from Cairo.
Israel returned his body on Monday and Egyptian authorities in turn handed it over to his family.
He was buried late on Monday with only his family attending the burial rites in Qalubiyah, a province just north of the Egyptian capital that is mostly considered part of Greater Cairo.
Israeli Prime Minister Benjamin Netanyahu described the incident as a terrorist attack and blamed the policeman.
“Israel relayed a clear message to the Egyptian government. We expect that the joint investigation will be exhaustive and thorough,” Mr Netanyahu told his cabinet on Sunday.
“We will refresh procedures and methods of operations and also the measures to reduce to a minimum the smuggling and to ensure tragic terrorist attacks like this do not happen again.”
On Sunday, Israel’s military chief ordered officials to conduct a preliminary investigation into the incident within a week.
Israeli media have raised a host of questions, including why two soldiers manning the guard post were not contacted for routine check-ups by radio for almost five hours. Other questions have included why the pair were posted on a 12-hour shift in the summer heat.
Local media released pictures showing the crossing point being secured only by zip ties, leading to questions about why the gate used by the policeman to cross the border was so lightly guarded.
According to reports, the Egyptian policeman was eventually spotted by a drone.
Egypt has neither confirmed nor denied that the policeman intentionally killed the Israeli soldiers. Officials, meanwhile, said authorities were fully co-operating with their Israeli counterparts.
Part of the investigation involved examining footage from security cameras installed at the area where the police officer pursued drug smugglers and fired at them.
Egyptian officials said the joint investigation was being carried out in a professional and friendly manner.
The two sides are also co-operating on ways to upgrade guidelines on the border to ensure Saturday's incident is not repeated, officials said.
Egypt and Israel co-operate closely on security and counterterrorism in the vast Sinai Peninsula, where the border is located. Egypt recently quelled an insurgency by militants in Sinai's north-east corner, close to the border with Israel and the Gaza Strip.
The two countries also co-ordinate to counter drugs and human trafficking.
Egypt is the main interlocutor between Israel and the militant Palestinian Hamas group, which rules the Gaza Strip, frequently mediating in a bid to end hostilities.
Hours before the shooting, Israeli soldiers had foiled an attempted drug smuggling operation at the border, seizing contraband estimated to be worth 1.5 million shekels ($400,000), an Israeli military spokesman said.
The spokesman said no link had been established between the drug seizure and the attack along the border.
In 1979, Egypt became the first Arab state to sign a peace treaty with Israel. The border between the two former foes has been mostly peaceful following the accord, although drug smugglers have occasionally exchanged fire with Israeli troops.
There have been incidents in the past on the border, none of which posed a serious threat to relations.
In 2011, assailants who came from Sinai killed eight Israelis in an ambush north of Israel's Red Sea port city of Eilat at the northern point of the Gulf of Aqaba. Pursuing Israeli forces killed seven of the attackers and five Egyptian policemen.
In 2012, three militants who infiltrated from Sinai were killed, as well as an Israeli soldier, in a clash along the border. In 2014, two Israeli soldiers on patrol were wounded by unidentified men who fired an anti-tank weapon from Egypt during a drug-smuggling attempt, according to the Israeli military.
In 2015, rockets fired from Sinai hit southern Israel but caused no casualties. ISIS later claimed responsibility.
Additional reporting by The National's Jerusalem correspondent, Thomas Helm