MOSCOW — Tens of thousands of Russians embraced the numbing cold and marched to a frozen riverbank near the Kremlin on Saturday, demonstrating their determination to keep up the pressure on Vladimir Putin for fair elections and honest government.

But this time, the Russian authorities were prepared, organizing a simultaneous, and also huge, rally in defense of Putin. Speakers issued dire warnings of the possible consequences of continued protest revolution and the breakup of the country.

By the end of the frigid day, it appeared that the anti-government demonstrations had not lost momentum and could continue into the spring. If they do, they will pose an unexpected challenge to Putin, who has never faced sustained public opposition in his 12 years as the country’s paramount leader.

City authorities said Saturday’s anti-government crowd was larger than either of the two similar rallies in December; they estimated about 36,000 people were there. Organizers gave an estimate of 120,000.

A troubled opposition

With precisely a month left before presidential elections, polls show that Putin, who is currently prime minister, is far ahead of his four rivals in the race. The protest movement has not coalesced into a coherent political force.

The opposition has also drawn some criticism for including communists and nationalists in its ranks. Separately from the main march, Saturday also saw a small Moscow rally by anti-Putin figures who want to keep their distance. They expected as many 30,000, but only about 250 came.

The rally ended with the call of “Not a Single Vote for Putin” and demands for legal reforms that would open the way for fair political competition and for new parliamentary and presidential elections. The protesters also demanded the release of political prisoners and punishment for those involved in the vote-rigging.

Before heading home, the protesters released white balloons. Some balloons had lettering saying “For Fair Elections” or “If You Inflate (the vote) Once Again, I’ll Burst.”