* Rocket destroyed near border with Egypt's Sinai
* Eilat packed with tourists; no injuries, damage reported
* Sinai-based Islamist group claims responsibility (Adds colour, edits)
By Allyn Fisher-Ilan
JERUSALEM, Aug 13 Israel shot down a rocket aimed at its southern resort city of Eilat early on Tuesday, blowing it up in mid air near the border with Egypt's Sinai Peninsula, a military official said.
Islamist militants operating in the lawless desert peninsula said they fired the rocket at the city, which was packed with tourists, in retaliation for the killing of four guerrillas in Sinai on Friday.
Air raid sirens rang out and blasts reverberated over the hills surrounding the resort on the shores of the Gulf of Aqaba, witnesses and Israeli news media said. No casualties or damage were reported.
A retired Israeli security official said it was the first time Israel's Iron Dome interceptor system had struck a missile fired at Eilat.
"It's not the first time that a rocket has been fired at Eilat, but it is the first time the Iron Dome has intercepted one," Dani Arditi, former national security adviser told Army Radio.
Israel briefly shut its airport in the city on Thursday citing threats from militants in Sinai involved in a standing confrontation with the Egyptian military.
Israel has also boosted its rocket defences near its southern border with Egypt.
Violence in Sinai picked up after former Egyptian president Hosni Mubarak, an ally of Israel, was toppled in 2011. It has increased again since last month's military overthrow of Mubarak's successor, Islamist president Mohamed Mursi.
Magles Shoura al-Mujahideen, a hardline Islamist faction which also has a small presence in the Gaza Strip, said it fired the missile at Eilat.
"In a swift response to the latest crime of the Jews that killed 4 Mujahideen in the land of Sinai through a strike by a pilotless plane, the lions of Maglis Shoura al-Mujahideen ... were able to strike the occupied city of Om al-Rashash "Eilat" with a rocket," it said in a statement published online.
"We assure that neither Eilat not any other Israeli cities will be blessed with security, tourism or economy," it added.
Eilat's mayor said there had been no panic.
"I think the military prepared in the best possible way with an enhanced defensive shield to protect Eilat and tourists including the (Iron Dome) battery... I and my citizens feel secure," Meir Yitzhak-Halevy told Army Radio.
Al Qaeda accused Israel of carrying out Friday's missile strike in Sinai but Israel dismissed the accusation. Egypt's armed forces said one of its helicopters fired the missile at al Qaeda fighters.
The Sinai was largely demilitarised as part of the Israeli-Egyptian peace treaty of 1979. But Israel has allowed Egypt to send troops to the area to combat the militants and arms smuggling by Palestinians into Gaza.
The peninsula has long been a security headache for Egypt and its neighbours. Large and mostly uninhabited, it also borders the Palestinian-ruled Gaza Strip and the Suez Canal linking Asia to Europe.
It is home to nomad clans disaffected with rule from Cairo. (Additional reporting by Shaimaa Fayed in Cairo and; Mahmoud Habboush in Dubai; Writing by Allyn Fisher-Ilan and Ori Lewis; Editing by Mohammad Zargham and Andrew Heavens)