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Šrū Caitanva-caritāmrta
[[Madhya-lilā Ch. 6]]
“‘However,’” the lord told Sanātana. “I shall explain the chief  lilā-avatāras .”
matsya, kūrma, raghunātha, nṛsiṁha, vāmana varāhādi—lekhā yāṅra nā yāya ganana
Thus the Lord's incarnations were enumerated, including Matsya, the fish incarnation; Kūrma, the tortoise; Lord Rāmacandra; Nṛṣiṁhadeva; Vāmanadeva; and Varāha, the boar incarnation. Thus there are innumerable  lilā-avatāras , and all of these exhibit wonderful pastimes. Lord Varāha, the boar incarnation, lifted the entire planet earth from the depths of the Garbhodaka Ocean. The tortoise incarnation, Lord Kūrma, became a pivot for the emulsification of the whole sea, and Lord Nṛṣiṁhadeva appeared as half-man, half-lion. These are some of the wonderful and uncommon features of  lilā-avatāras .
In his book  Laghu-bhāgavatāmṛta , Śrīla Rūpa Gosvāmī has enumerated the following twenty-five  līlā-avatāras : Catuḥ-sana, Nārada, Varāha, Matsya, Yajña, Nara-Nārāyaṇa, Kapila, Dattātreya, Hayaśīrṣa (Hayagrīva), Haṁsa, Pṛśnigarbha, Ṛṣabha, Pṛthu, Nṛsiṁha, Kūrma, Dhanvantari, Mohinī, Vāmana, Paraśurāma, Rāgha-vendra, Vyāsa, Balarāma, Krṣna, Buddha and Kalki.
Śrī Caitanya Mahāprabhu is not mentioned as a  lilā-avatāra  because He is an incarnation in disguise ( channa-avatāra ). In this age of Kali there are no  lilā-avatāras , but there is an incarnation of the Lord manifested in the body of Śrī Caitanya Mahāprabhu. This has been explained in  Śrīmad-Bhāgavatam .
TEXT 100
প্রতিযুগে করেন কৃষ্ণ যুগ-অনতার ।।
ଓର୍କନିଉର୍ଡ଼ ହାନ୍ସ   ତୋରାମ ନାହିଁକି ସିରାମ ॥ ১୦୦ ॥
pratiyuge karena krșṇa yuga-avatāra tarka-niṣṭha hṛdaya tomāra nāhika vicāra
SYNONYMS
prati-yuge —in every age or millennium;  karena —makes;  kṛṣṇa —Lord Kṛṣṇa;  yuga-avatāra —incarnation for the age;  tarka-niṣṭha —hardened by argument;  hṛdaya —heart;  tomāra —your;  nāhika —there is not;  vicāra —consideration.
TRANSLATION
Gopīnātha Ācārya continued: “There is certainly an incarnation in every age, and such an incarnation is called the yuga-avatāra. But your heart has become so hardened by logic and argument that you cannot consider all these facts.