Children celebrate on tank abandoned by  fleeing Iraqi troops in northern Iraqi town  of Tuz Khurmatu on Thursday. (CNN)

KIRKUK, Iraq - A convoy of Kurdish peshmerga fighters moved into the oil-rich city of Kirkuk on Thursday. The fighters from the Kurdish autonomous area drove through an industrial neighborhood of the city they have long dreamed of regaining from President Saddam Hussein's regime.  It was unclear if any Iraqi forces were still in Kirkuk, but there was some shooting on the northwest edge of the city.  A U.S. military official, speaking on condition of anonymity, said American special operations forces were attempting to get a U.S. presence into the city "in the interest of regional stability" - an apparent reference to Turkey's concerns about Kurds taking over the key oil city.

The cars and trucks flew the flags of the two major political parties who rule the two sectors of the autonomous region; yellow for Kurdistan Democratic Party and green for the Patriotic Union of Kurdistan. 

The Kurdish leadership has sworn to cooperate with U.S. military authorities in the campaign to win control of Kirkuk and the other major northern city still under government control, Mosul. 

Kirkuk, a predominantly Kurdish city, is the center of one of Iraq's richest oil regions. 

The fighters drove around at will Thursday. Kurdish families ran out of their homes to cheer and throw roses as the fighters passed. 

In a square, the bodies of two dead Iraqi soldiers lay near the base of a statue of Saddam. A chain was wrapped around the figure of Saddam, perhaps in preparation for toppling. 

People moved around, ignoring the Iraqis' bodies. 

Offices of the Iraqi government and the ruling Baath Party were being looted, with residents carrying away air conditioners and other goods. 

Along one road, a boy pushed a cart full of office chairs.