* King has undergone successful surgery, TV says
* Crown prince, also in his 80s, governing kingdom
(Adds television report, background)
RIYADH, Nov 24 (Reuters) - Saudi Arabia's elderly King Abdullah, who is in New York for medical treatment, has undergone a "successful operation", Saudi-owned Al Arabiya television said on Wednesday.
The king, thought to be around 86 or 87, arrived in New York on Monday to seek treatment after a blood clot complicated a slipped spinal disc.
Dubai-based Al Arabiya gave no further details. There was no immediate statement by the royal court.
A frail Crown Prince Sultan, who has health problems of his own, has returned from abroad to govern the world's largest oil exporter, while Abdullah is away for an unspecified period.
The kingdom is keen to show its allies in Washington and elsewhere there will be no power vacuum as health problems beset its octogenarian rulers, but the question of whether a reformist or a conservative will take over remains a matter of concern.
With both Abdullah and Sultan in their 80s, the next in line would be conservative Interior Minister Prince Nayef, at a relatively youthful 76. Nayef would still need the approval of Saudi Arabia's "Allegiance Council" to become king.
Abdullah appointed his half-brother Nayef second deputy prime minister in 2009 in a move that analysts say will secure the leadership in the event of serious health problems afflicting the king and crown prince and improve Nayef's chances of one day being king.
Diplomats in Riyadh say Western governments concerned about the fate of social and economic reforms promoted by Abdullah have reservations about the ascent of Nayef, seen as a religious and social traditionalist.
King Abdullah, who came to power in 2005, is the sixth leader of Saudi Arabia, whose political stability is of regional and global concern. It controls more than a fifth of the world's crude oil reserves, is a vital U.S. ally in the region, a major holder of dollar assets and home to the biggest Arab bourse.
As home to Islam's holiest sites, as well as birthplace of al Qaeda leader Osama bin Laden, Saudi Arabia is key to global efforts to fight Islamic militancy.
Washington wants Riyadh to continue social and economic reforms promoted by Abdullah that were seen as crucial after a group composed of mainly Saudis carried out the Sept. 11 attacks of 2001.
Diplomats say there has been uncertainty about Abdullah's health since he cancelled a visit to France in July.
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