Many connections can be found between the praṇava oṃ and the mantra “haṃsa” (sohaṃ). The symbol of the swan, like a bird, is found in the Nādabindūpaniṣad. This image and the connection with praṇava are described there as following:

अकारो दक्षिणः पक्ष उकारस्तूत्तरः स्मृतः।
मकारं पुच्छमित्याहुरर्धमात्रा तु मस्तकम् ॥मस्तकम् १॥

akāro dakṣiṇaḥ pakṣa ukārastūttaraḥ smṛtaḥ।
makāraṃ pucchamityāhurardhamātrā tu mastakam ॥mastakam 1॥

It is believed (smṛtaḥ) the mātrā “A” (akāraḥ) is in the right wing (dakṣinaḥ-pakṣaḥ), the mātrā “U”(ukāraḥ) in the north (uttaraḥ – “north” is “in the left” wing). Thus (iti), in his tail (puccham) is the mātrā “Ma” (makāraṃ), also (tu) they say (āhuḥ), ia half the mātrā, (ardhamātrā) is in his head (mastakam), i.e. nasalized vibration.