The world's largest church and the biggest of the Orthodox churches have stepped up contacts in recent years and inched towards holding the first summit of their leaders in history.
The late Pope John Paul used to read the Bible in Russian, preparing for a trip he never got to take.
Pope Benedict's aides are in regular contact with Moscow but Alexiy wanted some disputes solved before any meeting.
Metropolitan Kirill of Smolensk, the Orthodox church's top diplomat for the past two decades, is seen as the frontrunner to follow Alexiy, who died on Friday aged 79. He is considered open to cooperation with Catholics. But experts say his chances are uncertain, especially since the election method has not been decided on.
"Who will they choose? Who knows?" said Father Jivko Panev, theology lecturer at Saint Sergius, the Paris seminary of the Russian Orthodox Exarchate in Western Europe.
The Great Schism of 1054 split Christianity into eastern and western wings. Catholics number 1.1 billion, more than half the world's 2 billion Christians, while the Russians make up over half of the 220 million Orthodox around the globe.
While known and appreciated abroad, Panev said, Kirill does not have strong support among the bishops at home who will meet to elect the successor within the next six months.
USING THE "APOSTOLIC METHOD"?
The method of electing a successor still has to be decided by the church's Holy Synod and could bring further uncertainty. One option would have the bishops choose three candidates from their ranks and then pick the winner by lots.
Choosing by lots is known as the "apostolic method" because the original Apostles are believed to have used it to replace the traitor Judas. It is said to leave the final decision to the Holy Spirit, the third person of God in Christian theology.
The Russians have used this in the past, but not in 1990 to pick Alexiy, who was elected unopposed. "They still used Soviet methods then," said Canon Michael Bourdeaux, head of the Keston Institute that monitors religion in former communist states.
"The three would probably be Kirill, Clement of Kaluga and a third man," Panev said. Clement is a conservative prelate seen as a Russian nationalist wary of closer contacts to the West.
The main obstacle to better relations is the Russian charge that Rome has been trying to woo over Orthodox to Catholicism since the end of communism in 1991, a charge the Vatican denies.
"The Russian attitude is that they own Russia spiritually and the Catholics are poaching," said Professor John Anderson, an expert on Russian Orthodoxy at the University of St. Andrews in Scotland. This was not likely to change, he said.
But the Russians have sounded more positive about the Catholics recently, Anderson said: "They seem to be trying to put aside the differences to stress what they have in common."
Both Anderson and Panev said the majority of Russian bishops were more conservative than Alexiy and Kirill and shared a traditional Russian suspicion of the West, especially Catholics.
Alexiy mentioned this at a Synod of Bishops in June, saying that ultra-conservative bishops who opposed wider ecumenical ties had created "an almost open confrontation" in the church.
Despite this, he said that "a positive dialogue is being conducted with the Roman Catholic church on a whole series of pressing contemporary questions." He and Kirill favoured working with the Vatican to defend Christian values in Europe.
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(Source: JVN)