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In Greek mythology, the name Branchus (Greek: Βράγχος) refers to the following characters, who may or may not be identical:
Branchus, a son of Smycrus and lover of Apollo; his mother was a woman from Miletus, who, while she was pregnant with him, had a vision of the sun entering her mouth and reaching her belly. Hence when the child was born, he was named Branchus. When he was a young man, he came across Apollo in the forest and kissed him. Apollo embraced him in return and gave him a crown and a staff. Branchus, receiving prophetic abilities from Apollo, introduced the worship of the god at Didyma. He also founded a shrine at Miletus. His descendants, the Branchides, were an influential clan of prophets.