May 17 (Reuters) - Lebanon's rival leaders hold talks in Qatar on Saturday to try to end their political conflict.

The talks will be held according to the terms of a deal brokered by a Qatari-led Arab League delegation which halted the worst fighting among Lebanese since the 1975-90 civil war.

The talks in Doha will seek agreement on two points: how to share seats in a new cabinet and a new parliamentary election law.

Agreement on these issues would pave the way for parliament to convene to elect army commander General Michel Suleiman president. His nomination to the post, which has been vacant since November because of the crisis, is accepted by both sides.

Once Suleiman is elected, the new cabinet will be formed.

GOVERNMENT OF NATIONAL UNITY

The division of seats in cabinet has been a main point of dispute between the U.S.-backed ruling coalition and the Hezbollah-led opponents for 18 months.

The crisis began in November 2006, when the ruling coalition's refusal to give the opposition effective veto power in cabinet prompted the resignation of Shi'ite ministers from Hezbollah and the allied Amal movement. With no Shi'ite ministers, the opposition declared the cabinet illegitimate.

Government in Lebanon is run according to a delicate sectarian system of power-sharing.

The opposition launched a campaign of street protests in December 2006 to press its demands.

Various proposals have been floated for settling the dispute. The opposition has called for an equal three-way division of the cabinet. A third of the seats would go to the ruling coalition and a third to the opposition. The remaining third of the ministers would be appointed by the president.

This idea has so far been rejected by the ruling coalition.

PARLIAMENTARY ELECTION LAW

Lebanon's last parliamentary election was held in 2005 after the withdrawal of Syrian troops from the country and returned a majority for the anti-Syrian ruling alliance.

The election was held under a controversial law drawn up in 2000 when Syria dominated politics in the country.

The sides must agree on a law for the next parliamentary election scheduled for 2009. They are split over the size of electoral constituencies and the delineation of their boundaries.

Seats in the 128-seat parliament are divided equally between Christians and Muslims under the Taif Accord which ended the civil war. (Writing by Tom Perry in Beirut; Editing by Elizabeth Piper)

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