Her murder provoked international condemnation and critics accused Moscow of failing to protect freedom of speech.
Burayev was a former candidate for Chechnya's presidency and was head of the Achkhoi-Martan region from 1995 until he was sacked by the region's then-president, Akhmad Kadyrov, in 2003.
Last month, 10 other people were arrested in relation to the case, according to Russian news agencies.
Politkovskaya was active in exposing abuses by security forces in Chechnya and prosecutors said her murder could be linked to her reporting.
She was the 13th journalist killed in a contract-style killing in Russia since Vladimir Putin, the Russian president, took office in 2000, the US-based Committee to Protect Journalists said.
Nobody has been convicted in any of the cases.
Politkovskaya made regular trips to report on the Kremlin's operations against Chechen fighters and her reporting attracted the attention of officials and the security forces.
Putin said last December that everything possible was being done to find and punish her killers.
"The best professionals in Russia's law enforcement are investigating this crime and we already have certain results," he said.