The swan has a special association with advaita vedAnta. The swan is called hamsa in the sam.skRta language. The greatest masters in the advaita tradition are called paramahamsas - the great swans. The word hamsa is a variation of so'ham: I am He, which constitutes the highest realization. There are other equivalences between the swan and the advaitin, that make the swan a particularly apt symbol for advaita vedAnta. The swan stays in water, but its feathers remain dry. Similarly, the advaitin lives in the world, yet strives to remain unaffected by life's ups and downs. The swan is thus a symbol for the jIvanmukta, who is liberated while still alive in this world, by virtue of having realized Brahman.
