(Recasts, adds details, quotes throughout)
By Michelle Nichols
UNITED NATIONS, May 13 (Reuters) - Syria mediator Lakhdar Brahimi will step down on May 31, U.N. Secretary-General Ban Ki-moon said on Tuesday, blaming an international deadlock over how to end the country's three-year civil war for hampering a bid to broker peace.
International and regional powers have backed opposing sides in the civil war that has so far killed at least 150,000 people, with Russia and Iran supporting President Bashar al-Assad and Western powers and Gulf Arab states largely backing the rebels.
Brahimi had long threatened to quit, just as his predecessor - former U.N. Secretary-General Kofi Annan - did in 2012. Annan resigned after six months as the U.N. and Arab League joint special representative on Syria, slamming the U.N. Security Council for failing to unite behind efforts to end the fighting.
Brahimi, a veteran Algerian diplomat, briefed the 15-member Security Council for the final time on Tuesday.
"I go with a heavy heart because so little was achieved," he told the closed-door session of the council, according to the his statement which was obtained by Reuters. "I once again, humbly apologize to the Syrian people."
He left "a few ideas" with Ban and the Security Council - focus on getting more aid into Syria, reduce and end the violence, end the flow of arms into Syria and help form an executive to lead Syria's transition by organizing a national dialogue, conference, constitutional review and elections.
Ban said he would work on finding someone to replace Brahimi. There are several possible candidates, diplomatic sources have said, among them former Tunisian Foreign Minister Kamel Morjane.
"At this time, I have to think who should be the right person and at what time," Ban said.
Ban praised Brahimi's perseverance in the face of "impossible odds, a Syrian nation, Middle Eastern region and wider international community that have been hopelessly divided in their approaches to ending the conflict."
"That his efforts have not received effective support from the United Nations body that is charged with upholding peace and security, and from countries with influence on the Syrian situation, is a failure of all of us," Ban told reporters.
'SHOW MUST GO ON'
The 15-member U.N. Security Council has been split on how to deal with the war in Syria. Syrian ally Russia, supported by China, has blocked three resolutions that would have condemned President Bashar al-Assad's government, threatened sanctions and called for war crimes accountability.
That deadlock is set to be exposed again when the council votes on a French-drafted resolution to refer Syria's civil war to the International Criminal Court. French U.N. Ambassador Gerard Araud said he hoped a vote could happen early next week.
But Russian U.N. Ambassador Vitaly Churkin said on Tuesday of the bid to send Syria to The Hague-based court: "We don't think it's a good idea." He added that the move "definitely will have the effect of exacerbating those (council) differences."
In announcing Brahimi's departure, Ban said: "There must be accountability for the terrible crimes that have been - and are being - committed. Such crimes include the deliberate starvation of communities by preventing humanitarian access."
Australia, Luxembourg and Jordan are also drafting a second resolution to impose "measures," said Australian U.N. Ambassador Gary Quinlan, after the warring parties effectively ignored a resolution adopted by the Security Council in February that demanded greater humanitarian access, including across borders.
The Security Council could impose targeted sanctions on individuals violating international humanitarian law, although Russia is again unlikely to support such a move, diplomats say.
Russia proposed its own brief draft resolution on Tuesday on the humanitarian situation in Syria, which diplomats said urged the warring parties to replicate around the country a deal made in Homs that allowed rebels and civilians to be evacuated.
"Lakhdar Brahimi has worked so hard to try to accomplish a breakthrough in a political settlement in Syria, unfortunately it hasn't happened so far, but we believe that the show must go on ... to find a political solution to Syrian crisis," Churkin told reporters.
Syrian U.N. Ambassador Bashar Ja'afari said the Syrian government noted Brahimi's resignation, adding that he should not be blamed for "all of the mistakes."
"The American administration made mistakes, the French government, the British, the European Union, Qatar, Saudi Arabia, Turkey - all of them made mistakes," he said. "The main mistake was the interference by these foreign powers into the domestic affairs of the Syrian people."
U.S. Secretary of State John Kerry said Brahimi did not fail. "It is the fault of a party, Assad, who will not negotiate, who absolutely refused to negotiate at every single session," he told reporters in Washington.
Brahimi said a political settlement would be impossible without the involvement of all countries in the region. In January Ban withdrew an invitation to Iran to attend the first round of Syria peace talks under pressure from the United States and boycott threats by the Syrian opposition.
Brahimi told the Security Council about a four point plan proposed by Iran's Foreign Minister Mohammad Javad Zarif - a ceasefire, a national unity government, a constitutional review that would limit the powers of the president, and elections.
"Their ideas are interesting and I think probably will be considered," Brahimi told reporters after the council briefing.
Brahimi organized two rounds of negotiations in Switzerland between Assad's government and members of the opposition.
While there were no breakthroughs at those talks, diplomats and U.N. officials said Brahimi had wanted to continue the process to find a negotiated solution to end the fighting.
But Syria's announcement that it would hold presidential elections on June 3 dealt a severe blow to Brahimi's efforts.
"The announcement by the Syrian government that it intends to hold presidential elections next month, with the serving president as the only serious candidate, puts additional pressure on the prospect for a settlement," he told the Security Council.
The election is widely seen as a bid by Assad to defy widespread opposition and to extend his grip on power. (Additional reporting by Mirjam Donath; Editing by Chizu Nomiyama, Jan Paschal, Bernadette Baum and Eric Walsh)