Various Incarnation of Vyasa and Lord Shiva

During the 'Varah' Kalpa of the Seventh 'Manvantar' lord Vishnu illuminated all the three world by his divine presence. This seventh Manvantara consisted of four yugas which repeated themselves in a cyclic way for twelve times.

1. The first dwapar of this seventh manvantar saw the manifestation of lord Shiva for the welfare of the brahmins. When Kaliyuga arrived Lord Shiva again manifested himself along with goddess Shakti and was known as Mahamuni Shweta. Lord Brahma had the priviledge of becoming his disciple.
2. During the second dwapar, sage Vyas existed as Satya, Prajapati and Lord Shiva became famous as 'Sutar'. Lord Shiva in his incarnation of Sutra had many disciples among whom 'Dundubhi' was very famous.
3. During the third dwapar sage Vyas took his incarnation as Bhargava and lord Shiva became famous as Daman. Lord Shiva in his incarnation as Daman had four disciples among whom Vishoka was very famous. When Kaliyuga arrived after this third dwapar. Lord Shiva along with his Disciples helped Sage Vyas.
4. During the fourth Dwapar Sage Vyas took his incarnation as Angira and Lord Shiva as 'Suhotra'. Even in this incarnation Lord Shiva had four disciples among whom Sumukh was very famous. Lord Shiva along with his disciples helped Angira.
5. During the fifth dwapar sage Vyas took incarnation as Savita and Lord Shiva as 'Kanka' who was very famous for his tremendous austerities. Kanka had four disciples among whom Sanak was very famous.
6. During the sixth dwapar sage Vyas took incarnation as Mrityu and Lord Shiva as 'Lokakshi'. Lokakshi had four disciples among whom Sudhama was very prominent.
7. During the seventh dwapar sage Vyas manifested himself as Indra and Lord Shiva as Jaigisatya. Jaigisatya had four disciples among whom Saraswat was very prominent.
8. During the eighth dwapar sage Vyas took incarnation as Vashishth and Lord Shiva as Dadhivahan. Dadhivahan had four disciples among whom Kapil was very famous.
9. During the nineth dwapar sage Vyas took incarnation as Saraswat and Lord Shiva as 'Rishabh'. Lord Shiva in his incarnation as Rishabhdeva had four disciples among whom Parashar was very famous. 

Source Shiva Puran

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On being enquired by Lord Brahma about his various incarnations, Lord Shiva said ---

1. "I shall incarnate as Shwet-Mahamuni during first Dwapar Yuga of Vaivaswat manvantar. My abode shell is at Chhogal peak of Himalaya mountain. I shall have four disciples who will be proficient in the Vedas."
2. "I will incarnate as Sutar during second dwapar. At that time, Saddyojat Prajapati will be holding he post of Vyasa.I will have four sons- Dundubhi, Shataroopa, Richika and Keluma."
3. "During third dwapar and at the time when Sage Bhargav will be acting as a Vyasa, I will take incarnation as Daman. I will have four sons- Vikrosh, Vikesh, Vipash and Papnashak."
4. "During fourth dwapar and at the time when Sage Angira will be acting as a Vyasa, I will take incarnation as Suhotra. I will have four sons- Sumukh, Durmukh, Durdhar and Duratikram."
5. "I will take incarnation as Kank during fifth dwapar and at the time when Sage Savita will be fulfilling the obligations of Vyasa; I will have four sons- Sanak, Sanandan, Sanatan and Sanatkumar."
6. "During Sixth dwapar, I will take incarnation as Laugakshi. At that time Sage Mrityu will be holding the post of Vyasa. My sons will be Sudhama, Viraja, Shankh and Padraj."
7. "Sage Shataritu will act as Vyasa during Seventh dwapar. I will have four sons- Saraswat, Megh, Meghvah and Suvahan."
8. "Sage Vashishtha will be the Vyasa during eighth dwapar. I will take incarnation as Dadhivahan. My sons will be- Kapila, Asuri, Panch, Shikhomuni and Vashkal." 
9. "Sage Saraswat will act as Vyasa during nineth dwapar. I will take incarnation as Rishabh. I will have four sons- Parashar, Garg, Bhargav and Angiras." 
10. "The tenth dwapar will see Tripad fulfilling the duties of Vyasa. I will incarnate as Bhavita Muni. I will have four sons- Bala, Bandhu, Niramitra and Ketu."
11. "Sage Trivat will be the Vyasa during eleventh dwapar. I will take incarnation as Ugra. My sons will be- Lambodar, Lambaksha and Lambakesh." 
12. "Sage Shatateja will perform the duties of Vyasa during twelfth dwapar. I will have four sons- Sarvagya, Sambuddhi, Sadhya and Sarva."
13. "Dharma will be the Vyasa during thirteenth dwapar. I will take incarnation as Bali and live at the hermitage of the Balkhilya. I will have four sons- Sudhama, Kashyapa, Vashishtha and Viraja."

This way, Lord Shiva elaborated upon his various incarnations till the period of twenty-eight dwapar, which went as follows.

DWAPAR	VYASA	SHIVA	SHIVA'S SONS
14th	Tarakshu	Gautam	Atri, Devarudra, Shravana & Srivishthak
15th	Traiyyaruni	Vedshira	Kuni, Kunibahu, Kusharir & Kunetrak
16th	Deva	Gokarn	Kashyao, Shukracharya, Chyavn & Brihaspati
17th	Kritanjay	Guhavasi	Utathya, Vamadev, Mahayoga & Mahabal
18th	Ritanjay	Shikhani	Vachashrava,Richik,Shyavasva& Yatishwar
19th	Bhardwaj	Jatamali	Hiranyabh, Kaushal, Laungaksi & Kuthumi
20th	Gautam	Attahas	Sumantu, Varvari, Kavand & Kushi Kandar
21th	Vachashrava	Daruko	Ramapksoda, Bhayani, Ketuman & Gautam.
22th	Shushmayan	Langli Bheem	Dharmik, Mahayotri, Bhallavi, Madhupinga,Shwetu, Ketu & Kush
23th	Trinabindu	Kalanjar	Ushik, Vrihadashva, Deval & Kavirevach
24th	Riksha	Shooli	Salihotra, Agnivesh, Unasva & Sharadvasu.
25th	Shakti	Dandi-Mundishvar	Chhagal, Kundak, Kumbhand & Prawahaj
26th	Parashar	Sahishnu	Ulook, Viddyut, Shambuk & Ashwalayan.
27th	Jatuk Karnya	Somsharma	Akshapad, Kumar, Ulook & Vatsa.
28th	Dwaipayan	Lakuli	Kushik, Garg, Mitra & Kaurushya

Source Linga Purana

Reference
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