The implementation of the deal, backed by Turkey and Iran but opposed by Syria's opposition, began at midnight Friday. A spokesman for rebel groups who attended the talks in Kazakhstan expressed doubt Friday about the agreement, including with regard to Russia's ability to force Iran and the Syrian government to respect a truce with the rebels.
The airspace over the safe zones may also be closed for the planes of the US-led coalition, with Russia's Foreign Ministry saying the issue is being negotiated between the Russian and American militaries.
A rebel commander said the general level of violence was reduced, but added: "Regime attempts (to advance) in the Hama countryside continue".
The walkout and the comments underline the huge difficulties of implementing such a deal.
Syria's government has said that although it will abide by the agreement, it would continue fighting "terrorism" wherever it exists, parlance for most armed rebel groups fighting government troops.
A previous cease-fire agreement signed in Astana on December 30 helped reduce overall violence in Syria for several weeks but eventually collapsed. The deal also calls for refugees to be allowed to return to the safe zones and services and infrastructure to be restored.
The Syrian government is not a signatory, but its state news agency has said it supports the plan.
The deal was signed by Russia, Iran and Turkey on Thursday, said the report.
United Nations special envoy for Syria Staffan de Mistura said the agreement was important because it was being implemented by three countries who could make it work.
Syrian opposition forces have not signed on to the deal and walked out of the Kazakhstan talks in protest over previous violations by Assad forces and their Russian supporters.
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Several ceasefires have been agreed on since Syria's conflict broke out in 2011, but they have failed to permanently stem the fighting.
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But Lavrentiev seemed to sketch out a broader geographical no-fly zone for USA and coalition military planes.
The deal calls for the establishment of four "de- escalation" zones across stretches of eight Syrian provinces.
Mr Fomin said that there had been no bombing raids by Russian aviation in the four zones since 1 May.
According to the memorandum, any clashes between the government forces and opposition armed groups must stop within the zones.
The largest covers northern Idlib province, the opposition-held enclave near the Turkish border, and parts of neighboring Aleppo, Hama and Latakia provinces where fighting has been fierce. "All Syrian flights over these areas will cease".
"The operation of aviation in the de-escalation zones, especially of the forces of the worldwide coalition, is absolutely not envisaged, either with notification or without".
Even if the regime and its allies hold off on attacking rebel groups, the dominant presence of former Al Qaeda branch Jabhat Al Nusra, which now goes by the name Hayat Tahrir Al Sham, in the proposed safe zone areas is a threat to the agreement.
That suggests that the USA military would no longer be allowed to fly over a number of critical areas where it already conducts operations and that it would be barred from all of the most important areas contested by the government and rebels not affiliated with the Islamic State.
The United States was represented at the talks but was not part of the agreement signed by Russia, Iran and Turkey.
Such zones, also referred to as non-conflict or safe zones, are meant to be areas where civilians can live without being targeted by any party in Syria's war.