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* Just back from Cuba, brother says president recovering
* Says "necrophiliacs" spreading rumors over Chavez
* President not seen or heard from since Dec. 11 surgery
By Andrew Cawthorne and Diego Ore
CARACAS, Jan 12 (Reuters) - Venezuela's cancer-stricken president, Hugo Chavez, is recovering in Cuba and is not in a coma as some have rumored a month after surgery, his brother, Adan Chavez, said after a visit to Havana.
The 58-year-old socialist leader has not been seen or heard from since his Dec. 11 cancer surgery - his fourth such operation after the disease was detected in his pelvic area in mid-2011 - leaving Venezuela in a state of suspense.
But older brother Adan Chavez, who is governor of the family's home state of Barinas, said the president was improving daily, according to a news release on Saturday from his office.
"The head of state continues to assimilate treatment well and his recovery is advancing daily," the statement read.
"Information on social networks and in other places, saying the president is in a coma and his family are discussing the supposed disconnection of life support equipment, are totally false," it added.
Chavez missed his own inauguration for a new, six-year term last week, although Venezuela's top court ruled that he remained in power and Vice President Nicolas Maduro could deputize until there was clarity over the president's condition.
The rumors were stoked when Chavez did not send a message to Thursday's pro-government rally, the day he was supposed to be sworn in. Unlike past trips to Cuba for medical treatment, no images have been released of him.
The saga has enormous stakes for Venezuela, a nation of 29 million people with the world's largest oil reserves, as well as for the wider region. Cuba and a handful of other leftist-ruled nations depend on Chavez's economic aid.
Peruvian and Argentine Presidents Ollanta Humala and Cristina Fernandez, both friends of Chavez, visited Cuba this week. There was no sign either of them saw him.
Finishing her visit on Saturday, Fernandez said it was "inappropriate" for her to talk about Chavez's condition, which was a matter for his family. "I ask you to show a lot of respect and solidarity," she told reporters in Havana.
'DEAD OR ALIVE?'
Adan Chavez, a physicist by profession who has been a political mentor to his brother and is viewed by Venezuelans as a hard-liner, said foreign media were in league with local opposition activists to promote lies about the president.
"We know this is part of a dirty war by the necrophilic opposition," he was quoted as saying in the news release. "We are sure that with the support of God, science and the people, our president will triumph in this new battle."
Venezuela's opposition leaders are furious at what they see as a Cuban-inspired manipulation of the constitution by Maduro and other top "Chavista" figures aimed at preventing the naming of a caretaker president due to Chavez's absence.
Should Chavez die or have to step down, a new election would be called and would likely pit Maduro against opposition leader Henrique Capriles, the 40-year-old governor of Miranda state, who lost to Chavez in last year's presidential election.
He and other mainstream opposition leaders have criticized secrecy over Chavez's condition, but have taken a wait-and-see attitude, preferring to prepare behind the scenes for a possible new vote.
There have been small protests by students, none numbering more than several hundred people. A handful of people were injured in Tachira state on Friday, local media said, when protesting students clashed with police.
"Who knows if Chavez is alive or dead? They don't say clearly if he is breathing, if he can talk or not," 22-year-old university student Daniella Contreras said at a protest meeting in Caracas on Saturday.
"They should send a medical committee to Havana to confirm if the president is still capable of governing."
The government has been giving regular but terse updates on Chavez's condition, the latest being that he is struggling with a severe lung infection after the operation.
The silence from the normally garrulous leader famous for his lengthy speeches has led many Venezuelans to conclude his 14-year rule is ending.
Venezuela's most prominent female opposition activist, right-wing legislator Maria Corina Machado, told Saturday's gathering of about 400 protesters that the government was now illegitimate.
"We are in the terrible situation of having to acknowledge there is today no government in Venezuela. Government is in Cuba, led by the Cubans, deciding what we do and what happens with our country," Machado said.
Information Minister Ernesto Villegas urged Venezuelans to avoid being drawn into trouble. "An irresponsible minority are causing provocations to create a macabre show," he said.
The Chavez years have been turbulent ones, particularly during a short coup against him and a national oil strike in 2002 and 2003, and many Venezuelans are praying that whatever happens next, it will be non-violent.