What is the importance of Haṃsa - हंस - in Hinduism?

According to Sāyaṇa, Haṃsa does not indicate any physical creature, viz., swan.

Haṃsa emantes from han - to go, who goes eterenally (or) to destroy individually, ie., destruction of the feeling "I" or "aham";

Thus, realising that one is BRAHMAN;

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Haṃsa is identified in Rig Veda, with the Supreme Spirit, Ultimate Reality or Brahman.

ṛtam - ऋतम् can be identified with BRAHMAN.

Hamsa has been translated as swan or goose by some scholars.

Rig Veda 4.40.5

हंसः शुचिषद्वसुरन्तरिक्षसद्धोता वेदिषदतिथिर्दुरोणसत् । नृषद्वरसदृतसद्व्योमसदब्जा गोजा ऋतजा अद्रिजा ऋतम् ॥

haṃsaḥ śuciṣad vasur antarikṣasad dhotā vediṣad atithir duroṇasat | nṛṣad varasad ṛtasad vyomasad abjā gojā ṛtajā adrijā ṛtam ||

Sri Aurobindo’s Interpretation

This is the swan that dwells in the purity, the lord of substance in the middle world, the priest of the offering whose seat is upon the altar, the guest in the gated house. He dwells in the Man, he dwells in the Truth, he dwells in the wide Ether; he is born of the Waters, he is born of the Light, he is born of the Law, he is born of the Hill of Substance, he is the law of the Truth.

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