Machar gave no reason for the crash, but ruled out an attack.
Salva Kiir, southern vice-president, said: "Twenty-one passengers were killed and either two or three crew members.
"Two engines failed and there was nothing the pilot could do."
Luka Mariak, a spokesman for Kiir, said the accident appeared to be due to a mechanical failure.
However, officials in the government of South Sudan said that will continue to investigate the cause of the crash.
Gabriel Changson, a southern Sudan government spokesperson, said in a statement that the aircraft was "ancient", which may have results in the crash landing.
Attack ruled out
The bodies of the dead are now expected to be flown to the nearby United Nations Mission in Sudan (Unmis) airfield which is off-limits to the public.
Unmis has sent a helicopter to the flat, savannah-like area, the UN said.
"The plane had been rented from an charter company and was carrying a delegation of leaders from the Sudan People's Liberation Movement from Wau to the capital Juba," he said.
He said an inquiry would be launched.
The UN said the aircraft was a Beachcraft 1900 operated by South Sudan Air Connection traveling from Wau to Juba.
The former rebel SPLM signed an accord with the northern National Congress Party (NCP) in 2005, ending Africa's longest civil war.
Friday's crash comes a day after southern army officials said Sudan's northern and southern forces had agreed to withdraw from an oil-rich border, where clashes killed dozens last month.
Dim joined the national army in Khartoum in 1972 and rose to the rank of colonel before joining the SPLM's armed wing the Southern People's Liberation Army (SPLA) in 1987, according to the Sudan Tribune.
He left south Sudan to live in London in 1992, the paper said.
In July 2007, he was appointed defence minister and also minister for SPLA affairs, becoming the first to hold the post since the first southern government was formed after the war.