The general had initially assumed power, taking oath as the head of a military council that he said would rule the country for a two year-transitional period, but was forced to step down a day later in the face of mass protests for a civilian-led transitional administration.
An alliance of opposition parties and activists have vowed to continue protesting until the military council now in power heeds their demands.
Sudan's main protest group on Sunday demanded the immediate handover of power to a civilian transitional government, saying it would keep up the street demonstrations which ousted former President Omar al-Bashir last week to achieve its aims.
The army has appointed a military council that it says will rule for two years or less while elections are being organized.
In separate statements issued late Saturday, Saudi Arabia and the United Arab Emirates issued statements in support of Sudan's transitional military council.
Riyadh also announced a package of humanitarian aid, including petroleum products, wheat and medicines.
Burhan further vowed to "uproot" Bashir's regime and release protesters, in a bid to placate demonstrators demanding civilian rule.
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The military council and opposition parties are locked in talks in the capital Khartoum, while protestors continue a sit-in outside the military headquarters. He said that the talks would focus on submitting the organizers' demands and transition plan and that they are calling for dissolving al-Bashir's ruling National Congress Party.
Al-Bashir "set up an elaborate political-security system that only he was capable of running", and now they're "struggling with the conundrum of how to maintain consensus among a divided and militarized elite, and meet enough of the demands of the protesters to have a modicum of legitimacy". The protesters fear that the military, which is dominated by Al-Bashir appointees, will cling to power or select one of its own to succeed him.
"The ministry of foreign affairs is looking forward to the worldwide community to understand the situation and to support the transitional military council.in order to achieve the Sudanese goal of democratic transition", the ministry said in a statement.
Sudan has seen two leaders toppled in two days after General Awad Ibn Ouf, who ousted longtime ruler Omar al-Bashir, was himself ousted after 24 hours.
Al-Bashir imposed a state of emergency in February, banning unauthorised public gatherings and granting sweeping powers to the police in efforts to quash the unrest.
But while celebrating the fall of Bashir and then Ibn Ouf - a defence minister and long-time close aide of the deposed president - protesters remain cautious.
The Sudan Doctors Committee, an affiliate of the Sudanese Professionals Association, which has been spearheading the protests, said 13 people were shot dead on Thursday and three others, including the soldier, were killed Friday. "Our sit-in is the most powerful weapon in our hands", the SPA said in a tweet.