A day after President Obama’s special Middle East envoy called for a consolidation of the fragile Gaza cease-fire, the truce came under new strain on Thursday when the Israeli military said Palestinians fired a rocket into Israel at dawn and Israel launched an air attack into southern Gaza.

On his first visit to the region in his new role, the envoy, George J. Mitchell, traveled to the West Bank to meet with Palestinian leaders. On Wednesday, after discussions with Israel’s prime minister, Ehud Olmert, Mr. Mitchell said he spoke of “the critical importance” of consolidating the cease-fire that ended Israel’s three-week offensive against Hamas.

As Mr. Mitchell prepared to travel to Ramallah, Israel said it launched an air attack in the southern Gaza town of Khan Younis against a “known terrorist” accused by an Israeli military spokesman of being part of a squad responsible for a roadside bombing on Tuesday that killed an Israeli soldier on the Israeli side of the border.

News reports from Gaza described the target of the attack as a Hamas policeman on a motorcycle who was injured along with several civilians, including schoolchildren.

But the Israeli military spokesman, who spoke in return for customary anonymity, said the man was a member of a group called Global Jihad. The spokesman said the man had once been a supporter of Hamas, the Palestinian militant group that controls Gaza and that Israel holds responsible for all attacks from the coastal strip.

“As the sole authority in the Gaza Strip, Hamas bears full responsibility for all terrorist activity originating from Gaza,” an Israeli military statement said Thursday.

Global Jihad, a small and shadowy group that broke from Hamas, took responsibility for a roadside bombing on Tuesday. Israel retaliated with an airstrike that wounded a militant and a raid that killed a man whose family said he was a farmer.

On Wednesday, the Israeli military said a rocket, the first since the fighting ended on Jan. 18, was fired from Gaza hours after Mr. Mitchell arrived in Israel from Cairo. It landed in an open area in Israel, causing no injuries. Israel carried out a retaliatory air strike against what the military said was a weapons manufacturing plant in southern Gaza. There were no immediate reports of casualties.

Mr. Mitchell told reporters after the meeting with Mr. Olmert that a broadening of the truce should include a cessation of hostilities, an end to weapons smuggling into Gaza and “the reopening of the crossings” based on agreements reached in 2005.

Those agreements, brokered by the United States, called for Palestinian Authority forces loyal to President Mahmoud Abbas of Fatah, a Hamas rival, to secure the Palestinian side of the crossings. But Hamas took control of Gaza in 2007, routing the Palestinian Authority forces there. Israel has since imposed a strict economic embargo on Gaza, letting in only humanitarian aid and basic supplies.

An Olmert aide said the prime minister told Mr. Mitchell that the crossings would “not be permanently opened” until the case of a captured Israeli soldier, Cpl. Gilad Shalit, was resolved. Corporal Shalit was seized in a cross-border raid in 2006 and taken into Gaza. Hamas is demanding that Israel release hundreds of Palestinian prisoners, including many convicted of major terrorist acts, in exchange for his release.

Hamas has rejected any linkage between the reopening of the passages and the case of Corporal Shalit, and it insists on the reopening as a prerequisite to a lasting cease-fire. In a statement issued in Syria on Wednesday, the exiled leaders of Hamas and seven other Palestinian militant groups said the “factions of the resistance reject the signing of a truce agreement before the opening of all crossing points, the lifting of the blockade and the arrival of supplies.”

Mr. Mitchell planned to meet Mr. Abbas and other Palestinian Authority leaders on Thursday. Mr. Mitchell had no plans to meet with any representatives of Hamas, which the United States, like Israel and the European Union, classifies as a terrorist organization.

In Davos, Switzerland, meanwhile, the United Nations Secretary General, Ban Ki-moon, launched an appeal for $613 million in emergency aid for Palestinians in Gaza, saying: “Help is needed urgently,” news reports said.

Mr. Ban visited Gaza after both sides declared unilateral cease-fires almost two weeks ago. He is the highest-ranking international figure to have visited Gaza since the war. Mr. Ban was speaking to reporters covering the World Economic Forum in Davos.