* Putin mixes personal life, politics in third Kremlin term
* Admission of Kremlin split shifts focus to future
* Putin and Lyudmila rarely together in recent years
* Spokesman plays down talk of other relationship
By Steve Gutterman
MOSCOW, June 7 (Reuters) - Hours after President Vladimir Putin and his wife, Lyudmila, told Russians their marriage was over, an anchorman on a satirical on-line show gravely announced the latest news: Putin has named Lyudmila "acting First Lady".
It was a joke, of course, but Russians say there's a touch of truth in every joke: While the reasons for the separation may be deeply personal, the staged admission of a long-suspected estrangement was Putin's latest pragmatic political gambit.
The announcement was a piece of damage control, meant to tie up a loose end that has clouded the former KGB spy's image and allow him to tackle the six-year third term he started in May 2012 with a clean slate.
"It became necessary to resolve this situation - there were too many rumours swirling around, including that he has sent his wife to a convent," said Alexei Malashenko of the Carnegie Moscow Center.
Putin has mixed personal ties with politics in 13 years in power, seeking to strengthen his grip by shuffling his allies around and pushing foes from his path.
Now his wife of 30 years, long absent from his side, is officially out of the picture - almost like a Soviet apparatchik airbrushed from photographs after falling into disfavour.
Putin has championed family values and held out the Russian Orthodox Church as a moral guide for a society he says was set dangerously adrift by the Soviet collapse.
But the rhetoric has rung hollow against a backdrop of speculation he had abandoned Lyudmila, 55, for Olympic rhythmic gymnast Alina Kabayeva, 30, and fathered her child.
Blogger Leonid Volkov believes Putin wanted to erase the image of an unfaithful husband: "I've heard taxi divers say it many times: 'If he's cheating on his wife, it means he's deceiving the country,'" he wrote on Twitter.
Some of Putin's foes praised the decision. Opposition politician Boris Nemtsov said it was a rare show of honesty.
"Political life in Russia just became a bit more human," media magnate and Putin critic Alexander Lebedev tweeted.
Divorce is widespread in Russia, where many people marry young and the upheaval that followed the collapse of Communism has strained the social fabric.
"When I heard it over the television I was in shock," said Lyubov Andreyeva, a retired teacher in Moscow. "But at the same time if there is no feeling left and the children are grown up, what is the point of continuing the game?"
"I think the public will understand this decision and Putin will become more popular because everyone will see he is a regular, normal man, he can have family problems, like anybody else," said Malashenko.
But some Russians said the admission came too late.
"I think the time to do this was long ago because everyone already more or less knew about poor Lyuda, that she had gone insane or been sent to a convent or died," said Yulia Tsoy, a Muscovite. "It was senseless to continue that circus."
No Soviet leader or Russian president has divorced, and pro-Kremlin pundits cast the announcement - which the Putins made to a reporter after visiting the ballet in a rare public outing together - as courageous.
Irina, a 45-year-old homemaker in the Moscow suburbs who did not want her last name used, had a different word: "repulsive".
"This was done in a tacky way: 'Oh, the ballet was beautiful and I'm dumping my wife'", she said.
Irina is among Russians who are left cold by the stunts Putin has used to cultivate a manly image, from riding bareback without a shirt to stalking a tiger in Siberia in the name of wildlife preservation.
"I don't care about anything except how he governs," said another Irina, 63, in St. Petersburg, Putin's home town. "But he is the first of our leaders to allow himself this ... What I want to know is how she stood him for so many years."
Putin's job approval rating is lower than during his first two terms as president and fell to a 12-year low of 62 percent in January, according to the independent Levada Center.
"One of the most loyal segments of the electorate - married women - may now waver," political analyst Boris Mezhuyev wrote in Izvestia, a pro-Kremlin newspaper. "Putin has taken a quite risky and courageous step."
While the awkward joint appearance before a state TV camera late on Thursday confirmed what most Russians already assumed, it left the most titillating questions unanswered: Is Putin seeing another woman - or women - and will he marry?
For years, speculation has focused on Alina Kabayeva, a 2004 Olympic gold medallist who is now a parliamentary deputy for the Kremlin-backed United Russia party. Kabayev was the subject of a flattering documentary on state television in late May.
In 2008, at a news conference in Sardinia with then-Italian Prime Minister Silvio Berlusconi, Putin said there was no truth to a report he was to marry Kabayeva. He told journalists to keep their "snotty noses and erotic fantasies" to themselves.
In a poll on website peoplepassion.ru on Friday, about half the respondents clicked a box to say Kabayeva would be Putin's next wife, while almost as many said he would not remarry.
Ekho Moskvy radio cited spokesman Dmitry Peskov as saying there was no other woman in Putin's life and that he was too busy governing to have a family life. State TV said the couple have not divorced and that Putin still wore his wedding ring.
Tatyana Tsoy, a 52-year-old nanny, said the separation statement had clouded her image of Putin.
"I think his authority is already waning," she said. (Additional reporting by Thomas Grove, Gabriela Baczynska and Douglas Busvine)