Israel announced it would carry out an "exhaustive and thorough" investigation into a rare incident which left three Israeli soldiers killed on the border with Egypt, shortly after burying them.
Egypt has blamed the clash in the Negev desert on a member of its security forces who was also killed, saying he had crossed the border in pursuit of drug traffickers when he encountered the Israeli troops.
The Israeli army said an Egyptian "assailant" shot dead two soldiers at a post on the border. The discovery of their bodies triggered a manhunt during which the third soldier was killed as well as the suspected attacker.
Israel's border area with Egypt has not often witnessed violent incidents since Egypt made a peace deal Israel following the Camp David accords of 1978. Despite this, much of the country has taken an anti-Israeli stance due to its illegal occupation of the West Bank and the besiegement of the Gaza Strip.
Prime Minister Benjamin Netanyahu said it had sent Egypt a "clear message", speaking at the opening of a cabinet meeting on Sunday.
"We expect that the joint investigation will be exhaustive and thorough," he said.
"This is part of the important security cooperation between us, which has benefitted both countries over the years."
Egypt's army said a member of its security forces had crossed the border "chasing drug traffickers" before he was killed in an "exchange of fire which left three dead on the Israeli side".
The two Israeli soldiers whose bodies were found early Saturday at a guard post close to the Harif military base were identified as Lia Ben Nun, 19, and Ori Izhak Iluz, 20.
Ohad Dahan, also 20, was killed during the manhunt later on Saturday, the army said.
A fourth Israeli soldier, a non-commissioned officer, was lightly wounded, it added.
Their funerals took place in the Israeli cities of Rishon Lezion, Safed and Ofaim.
Meanwhile, the Egyptian policeman was identified as Sleiman Khater by Egyptian media, who said that Israeli troops killed him after "being found inside Israeli territory".
Khater's actions was largely lauded by Egyptians on social media, who hailed him as a 'martyr'.
Israeli media have raised questions over the shootings, particularly on how the assailant managed to cross the high barrier running along the border.
Netanyahu on Saturday promised a "full investigation" into the deaths, and Israeli senior government figures stressed the importance of cooperation with Egypt.
The Israeli army was conducting "a thorough investigation in collaboration with the Egyptian Armed Forces", said the military chief of staff Herzi Halevi.
Defence Minister Yoav Gallant highlighted "the importance of the ties between the two countries" following a telephone call with his Egyptian counterpart Mohamed Ahmed Zaki.
Zaki meanwhile underlined "the joint coordination to take the necessary measures to avoid the repetition of incidents of this kind in the future", according to a spokesman for the Egyptian army.
The border between the two countries is generally calm but has been the scene of regular smuggling attempts.
In recent years, there have been exchanges of fire between smugglers and Israeli soldiers stationed along the border.
In 2014, two Israeli soldiers on patrol were wounded by unidentified men who fired an anti-tank weapon from the Sinai during an attempt to smuggle drugs.