What is the name of the Phaeacian Queen in whose hands Odysseus carefully places a "two-handled cup"?

The name of the Phaeacian Queen is Aretes

Explanation:

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Arete, the Queen of Scheria. Alongside her husband (King Alcinous), she ruled the Phaeacians.

In the Odyssey, she appears as a noble and active superintendent of the household of her husband. When Odysseus arrived in Scheria, he appealed first to Arete for reception and protection, and she treated him hospitably.

Depicted in 'The Odyssey', this was a region in Greek mythology that was the last destination of Odysseus in his 10-year journey. When Odysseus arrived there, he prayed first to Arete for protection on Athena's suggestion. Athena protected him and treated him well