* Sudan accuses Juba of supporting SPLM-North rebels
* Talks ended without a breakthrough on Tuesday
* Both sides under pressure from oil shutdown (Adds South Sudan comments)
By Alexander Dziadosz and Charlton Doki
KHARTOUM/JUBA, Dec 19 (Reuters) - Sudan and South Sudan will tackle the sensitive issue of support for rebel groups for the first time when they resume security talks next month, Sudan's defence minister said on Wednesday.
The former civil war foes have been at loggerheads over their contested border and other issues since South Sudan seceded last year under a 2005 peace deal.
The disputes came to a head in January - when landlocked South Sudan shut down its entire 350,000 barrel-a-day oil output in a row with Khartoum over transit and other fees - and again in April, when border clashes brought the two close to all-out war.
Sudan accuses South Sudan of supporting rebels fighting in two states that border South Sudan. Juba denies supporting the rebels, known as the SPLM-North, and in turn accuses Khartoum of backing insurgents on its territory.
The SPLM-North rebels were part of the southern insurgent army during the civil war but were left in Sudan after partition.
Facing heavy pressure from the United Nations and African Union mediators, Sudan and South Sudan signed a raft of deals in September including one to pull back their armies from the border. Both sides say that step is needed to resume oil flows.
But talks in Addis Ababa about how to put those agreements into practice ended without agreement on Tuesday, and South Sudan's chief negotiator told Reuters the negotiations were now "deadlocked".
On Wednesday, Amum told reporters Khartoum was blocking implementation of the agreement by imposing new conditions, including a demand that South Sudan disarm the SPLM-North. Sudan has never confirmed or denied this.
"The preconditions Sudan has imposed for the implementation of the cooperation agreement and the resumption of oil production are unacceptable to us," he told reporters in Juba.
ECONOMIES UNDER STRAIN
Sudan's Defence Minister Abdel Raheem Mohammed Hussein said the negotiations had been "extensive" and that South Sudan had agreed to discuss its alleged support for Sudanese rebels.
"This is a big step," he told reporters at the Khartoum airport after returning from Ethiopia, adding that the two sides would meet again on Jan. 13 in Addis Ababa.
Hussein said the issue of rebel support had gone nowhere in previous meetings between security officials but there had been more progress this time.
"This time there was an agreement that this is an essential issue and should be discussed," he said, adding the two sides would discuss it during their next meeting.
Late on Tuesday, Thabo Mbeki, the African Union mediator and former South African president, said the two sides had agreed to take "practical steps" to demilitarise the border.
But the two were still in disagreement about how to dimilitarise a disputed strip of land known as Mile 14, he said.
"The parties continue to agree that it should be demilitarised but they are discussing how to effect that decision," Mbeki told reporters in Addis Ababa.
Both economies are facing severe strain because of the loss of oil revenues, which accounted for 98 percent of South Sudan's state income before the shutdown and were Sudan's biggest source of revenues and foreign currency before secession.
About 2 million people died in Sudan's north-south civil war, which left South Sudan economically devastated and awash with guns. (Additional reporting by Aaron Maasho in Addis Ababa; Editing by Alison Williams)