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The caretaker government loyal to ousted prime minister Yingluck Shinawatra is clinging to power and to the hope of an election in July to restore its authority.
But the government's enemies deride its legitimacy and are calling on the upper house of parliament, the courts and the Election Commission to appoint a new prime minister.
The head of the government team overseeing security during months of demonstrations against Yingluck and her brother, ousted former premier Thaksin Shinawatra, said protest leader Suthep Thaugsuban's call for a new prime minister was illegal.
"We would like to warn all Thais to stay away from the protest sites as we have to tighten our security forces in a bid to avert a crisis," Tharit Pengdit, chief of the Department of Special Investigation, told reporters.
Protesters have used guns and grenades to resist police efforts to clear them off the streets and the government has generally sought to avoid confrontation.
But Tharit's warning could be a sign that the government is feeling increasingly embattled, especially after Yingluck's sacking by the Constitutional Court for nepotism on Wednesday, and is trying to assert its authority.
The sometimes violent protests against Yingluck and Thaksin have sapped investor confidence, frightened off tourists and dented growth in Southeast Asia's second-biggest economy.
A day after Yingluck and nine of her cabinet members were thrown out of office she was indicted by an anti-corruption agency for negligence over a rice subsidy scheme that ran up big losses. The Senate is expected to impeach her for that, which could result in a ban from politics.
But Yingluck's Puea Thai party still runs the caretaker government and it is hoping to organise an election, tentatively scheduled for July 20, that it would probably win.
Thaksin or his loyalists have won every election since 2001. But his enemies say he is corrupt and buys votes and they want an appointed "people's council" to oversee electoral rule changes to stop the Shinawatras from winning.
Thailand has been divided for years by a struggle between the royalist establishment and Thaksin, a former telecommunications tycoon who fuelled a spectacular political rise with policies that won over the rural and urban poor.
But Thaksin's success posed a challenge to the traditional Bangkok-based power elite and he was dogged by accusations of corruption. He was ousted in a military coup in 2006 and has lived abroad since being sentenced to jail for graft in 2008.
The rival supporters are staging sit-in protests at various places in and on the outskirts of Bangkok, raising fears of violence.
Two anti-government protesters were injured in a suspected grenade blast on Saturday night outside the prime minister's offices, which have been vacant for weeks.
Yingluck's "red shirt" supporters have denounced her removal as a judicial coup and have warned of a tough reaction if their caretaker government is also thrown out.
The Senate is due to hold a special session on Monday to discuss the crisis.
The army, which has staged numerous coups since the end of absolute monarchy in 1932, has stayed out of the turmoil, with the army chief insisting that politicians have to settle the dispute. But substantial violence would raise the possibility of military intervention.