Moscow (CNN) -- A close ally of Russian Prime Minister Vladimir Putin is stepping down as the speaker of the lower house of Russia's parliament, their political party announced Wednesday.
Boris Gryzlov will retain his executive position in Putin's United Russia party, the party said on its website.
The resignation comes after allegations of fraud in parliamentary elections earlier this month led to the largest anti-government demonstrations Russia has seen in two decades.
Gryzlov said he had achieved much of what he set out to do in his eight years in the Duma, as the lower house is called, and will continue to head the political party's supreme council.
He added that he is "ready to accept a post determined by the President."
United Russia party officials said on their website they will nominate their candidate to replace Gryzlov next Saturday.
Tens of thousands turned out Saturday to protest against election results that returned Putin's United Russia party to power.
Police estimated crowds in Moscow at 25,000, while organizers said it was 40,000, the state-run RIA Novosti news agency reported. Either figure would make the protests the largest in the Russian capital in 20 years.
Claiming the results of parliamentary elections were rigged, protesters chanted "Putin out."
They also braved freezing temperatures in other Russian cities to demonstrate against what they said was vote fraud.
Another protest is scheduled for December 24 -- three days after the new parliament is due to hold its first session, under a decree signed Tuesday.
A day after the protests, President Dmitry Medvedev called for a probe into the allegations.
"I agree neither with the slogans nor the statements voiced at the protests," the statement on Medvedev's Facebook page said. "Nevertheless, I have ordered checks into all the reports from polling stations, regarding the compliance with the election laws."
Critics piled ridicule on the Facebook post.
"It's awful," Elena Panina wrote. "Really. He was drunk when he wrote that or he didn't read what he wrote. ... What slogans does our president disagree with? 'We are for fair elections?'"
Pro-government demonstrators took to the streets on Monday, but the rally was smaller than the weekend protests.
Also on Monday, Russian billionaire Mikhail Prokhorov -- the owner of the New Jersey Nets basketball team and one of the world's richest men -- said he will run for president of Russia next year.
Calling it "probably the most important decision of my life," he acknowledged the risks of challenging Russia's rulers.
"There is a saying in Russia: Never say never, anyone can end up behind bars. But I am not afraid," he said in a news conference in Moscow.
Prokhorov, 45, is worth $18 billion, Forbes estimated in March, making him Russia's third richest man.
His announcement set off a frenzy of speculation about whether he was running with the Kremlin's tacit backing, to give the impression there was a genuine contest for president, or whether he really seeks to defeat Putin, who also says he will run.
CNN's Alla Eshchenko contributed to this report.