Hamsa is the Swan sacred to Saraswati and Brahma. It is said to be a creature of great discriminating wisdom. I f you were to mix water and milk thoroughly in a bowl, and offer it, Hamsa would be able to put its beak in the bowl, and drink only the milk! This symbolizes the ability of pure consciousness to discriminate between the real and the unreal, between wisdom, and folly.
Because of their Sattvic nature, and their association with wisdom and learning, Swans are often exchanged between students and teachers as gifts.