अड्गार्ग्यगालवयोः

Operational rule vidhi

A सार्वधातुक-पित्-अपृक्त-हल्-प्रत्यय that follows the verbs रुद्, स्वप् , श्वस्, प्रण्, जक्ष् optionally gets an अट् आगम.

Word by word (padaccheda)

Substitute / term अट् prathamā · ekavacanaTarget गार्ग्यगालवयोः ṣaṣṭhī · dvivacana

Substitute / term Prathamā (1st case): in operational rules, the substitute or the thing introduced; in definitions, the term or its referent.
Target Ṣaṣṭhī (6th case): names what gets replaced, "in place of X" (1.1.49 षष्ठी स्थानेयोगा).

Dotted words are technical terms, dashed ones pratyāhāras (names for classes of sounds or affixes). Hover or tap one to see its card.

Full reading

अट् गार्ग्यगालवयोः

Pāṇini leaves out words that carry over from earlier sūtras. Expand to see the rule as a reader actually understands it.

As a rewrite rule (sketch from case endings)

गार्ग्यगालवयोः → अट्

Replace gārgyagālavayoḥ with aṭ. Inherited words (anuvṛtti) may add further conditions.

Commentary

According to the opinion of गार्ग्य and गालाव , the augment अट् comes before a सार्वधातुक affix consisting of single consonant, after the above five roots रुद् etc.
As अरोदत्, अरोदः, अस्वपत्, अस्वपः, अश्वसत् अश्वसः, प्राणत् प्राणः, अजक्षत्, अजक्षः ॥ The names of Gargya and Galava are mentioned for honoris causa.
The mention of these names is not for the sake of "option" (विकल्पार्थम्). Because the very injunction about अट्, would make the ईट् of the preceding sutra optional. The mention of more than one Acharya in the sutra is also for this very reason.

Śrīśa Chandra Vasu, The Ashtadhyayi of Panini (1897), via ashtadhyayi.com.