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* One mayor says migration crisis too big for Libya to handle

* Italy-Libya deal foresees funding for holding centres

* Around 181,000 arrived in Italy in 2016

By Ahmed Elumami

TRIPOLI, Feb 10 (Reuters) - Mayors from Libya's desert south to northern shores fear a deal struck between Tripoli and Rome to fund migrant holding centres in the north African country will simply shift Europe's migration crisis onto Libyan soil.

The Mediterranean sea between Libya and Italy has become the main crossing point for asylum seekers and economic migrants seeking a better life in Europe. Last year Italy recorded its record number of migrant arrivals.

The deal foresees European Union money for holding centres in towns and cities along the main human trafficking routes criss-crossing Libya, as well as training and equipment to fight the smugglers.

"Our priority is to support our own sons instead of allowing for illegal migrants in centres," said Hamed Al-Khyali, mayor of the southern city of Sabha, a migrant smuggling hub.

"If the Europeans want to allow them to stay, they can have them in their own lands, which are larger, but not in Libya because we have our own problems to take care of."

Libya descended into chaos after the 2011 toppling of long-time leader Muammar Gaddafi, enabling smuggling gangs to develop entrenched networks. They typically demand thousands of dollars from migrants for a risky journey across the desert before cramming them onto ill-equipped boats for a perilous crossing of the Mediterranean. An estimated 4,500 migrants drowned in 2016.

The agreement will depend heavily on the cooperation of local authorities along the smuggling routes because the U.N.-backed government in Tripoli exerts little effective control over much of the country.

Several mayors said they were not notified of the accord before it was struck.

The agreement, which has the backing of EU leaders, pledges support for "reception camps" where migrants can be held "until their deportation or their voluntary return to their countries of origin".

Some migrant detention centres already exist in Libya. A U.N. report in December said migrants in Libya were exposed to widespread abuse in the centres, which are generally controlled by armed groups though some have official status. The report also said some local officials were collaborating with the smugglers.

'DANGEROUS STEP'

Hussein Thwadi, mayor of the western coastal city of Sabratha, the departure point most frequently used for Mediterranean crossings by smugglers in Libya right now, said keeping migrants in Libya would be a "dangerous step".

"The idea of allowing illegal migrants to stay in Libya and providing good conditions for their livelihood is rejected by Libyans and by the authorities too," Thwadi said.

The migrant crisis was too great for Libyan authorities to handle, the mayor said.

"The problem of illegal migration must be solved internationally."

Most migrant-smuggling boats launch from western Libya. But the authorities in eastern Libya, which oppose the U.N.-backed government and control swathes of the south used by the human traffickers, this week rejected the Italian-Libyan deal.

Italian Foreign Minister Angelino Alfano on Thursday said this came as no surprise.

"We got ourselves a good deal, but it's not a magic wand, it doesn't mean that tomorrow morning all the problems will be resolved," Alfano told reporters.

This week the EU said it would try to protect migrants in Libya and increase voluntary repatriations through closer cooperation with the U.N. refugee agency and the International Organisation for Migration.

Both agencies have said that Libya should not be considered a safe country to hold migrants and process asylum requests. (Additional reporting by Antonella Cinelli; Writing by Aidan Lewis; Editing by Richard Lough)