By Michelle Nichols

UNITED NATIONS, Sept 30 (Reuters) - South Sudan's president and a rebel leader accused each other on Tuesday of violating a ceasefire brokered to end a 21-month conflict in the world's newest state as United Nations chief Ban Ki-moon appealed to them "not to betray and disappoint us."

Ban told President Salva Kiir and rebel leader Riek Machar that now was the time to correct their "grave mistakes." A political dispute between the pair spiraled into a war that has killed thousands and forced two million people to flee.

"We are all here to help you, I hope you will not betray and disappoint us," Ban told a meeting on the sidelines of the annual United Nations General Assembly in support of a peace deal signed last month.

Kiir has been president since independence from Sudan in 2011, and Machar was his deputy until he was dismissed in 2013.

The conflict reopened ethnic fault lines that pitted Kiir's Dinka against Machar's ethnic Nuer forces. Nearly 13,000 U.N. peacekeepers are still sheltering more than 200,000 people at camps throughout South Sudan.

"I am determined to stop this senseless war," Kiir told the meeting via video conference from South Sudan. "Unfortunately, while we are busy implementing the agreement, the armed opposition continues to violate the permanent ceasefire."

"When I decided to join the liberation struggle three decades ago I did not expect to liberate my people in order to take them back to war amongst themselves," he said.

Machar, who attended Ban's meeting in person, said he agreed with Kiir that the ceasefire was not holding. He said there were serious challenges to implementing the agreement signed last month, under mounting pressure from neighboring countries and threats of sanctions from Western powers and the United Nations.

"The strange thing is we are attacked, we are blamed (for violating the ceasefire)," Machar said, adding in reference to Kiir: "He hit us, than he ran to complain."

The United Nations has accused both sides of crimes against civilians. In June, the world body accused South Sudanese troops and allies of sexually abusing women and girls and reportedly burning some of them alive in their homes.

The United States and other Western donors have accused Kiir and Machar of squandering goodwill after South Sudan's independence and hindering development in an oil-producing nation with almost no tarmac roads and heavily reliant on aid. (Reporting by Michelle Nichols; Editing by Grant McCool)