* Socialist president out of sight during cancer treatment
* Foes say VP Maduro and others lying about his state
By Girish Gupta
CARACAS, March 3 (Reuters) - Hundreds of pro-opposition students and other critics of Hugo Chavez's government marched in Caracas on Sunday to demand proof that the cancer-stricken Venezuelan leader is still alive and governing.
The crowd, including various leaders at the more militant end of the Democratic Unity opposition coalition, sang protest songs and waved banners as they rallied in a central neighborhood on a sweltering morning.
"Give us the truth!" and "Stop lying!" read banners.
Underlining the deep political polarization of the South American nation of 29 million people during Chavez's 14-year rule, pro-government students were also planning a march in support of the president and his ministers.
With Chavez unseen, apart from one set of photos, since a Dec. 11 cancer operation in December, Venezuelans are on edge waiting for developments amid a sea of rumors.
Officials say Chavez is in a Caracas military hospital after returning from Cuba two weeks ago, battling for his life. Though he is breathing via a tracheal tube, unable to speak, and undergoing chemotherapy, the president continues to rule via written and other communications, they say.
Opponents, though, accuse Vice President Nicolas Maduro and others of lying about Chavez's condition. And there have been media and Internet accounts that Chavez may have died - all emphatically denied by the government.
"They are violating the constitution. Venezuela has no authorities right now. President Chavez is sick, he hasn't said a word in two months. He cannot govern," said protester Maria Montero, a 56-year-old teacher.
"We want impartial spokesmen to give us information about Chavez, real doctors, not politicians," added Maria Mendoza, 54, who works for state oil company PDVSA.
Should Chavez die or step down, a vote would be held within 30 days, probably pitting Maduro against opposition leader and state governor Henrique Capriles for leadership of a nation that holds the world's biggest oil reserves.
The stakes are also high for the region. Chavez has been the most strident Latin American critic of the United States and financed hefty aid programs for leftist governments from Cuba to Bolivia.
Capriles, who lost to Chavez in last year's presidential election, was on a private visit to the United States over the weekend, prompting accusations from the government that he was plotting and taking money from "fascist" factions there.
Calling him the "decadent prince of the parasitical bourgeoisie," Maduro said late on Saturday that Capriles had met with conservative U.S. political figures Otto Reich and Roger Noriega, and was also planning to meet a U.S. State Department official.
"We know very well about the conspiracies of these people. Let nobody be mistaken about Venezuela. Be careful, those who go abroad to conspire against our people," he said in one of a series of live addresses to the nation.
The opposition leader, a centrist politician who admires Brazil's model, retorted with a Tweet showing a photo of him with two young boys he said were his nephews.
"Here I am with two big conspirators," Capriles joked.
Sunday's demonstration was called to support several dozen students who have chained themselves up in a Caracas street for nearly a week to protest against the secrecy over Chavez.
The government has warned protesters against trying to approach the hospital where Chavez is being treated on a closely guarded ninth floor. (Writing by Andrew Cawthorne; Editing by Will Dunham)