Section IV
 1. Dhanwantari Gothra 2. Jamadagni Gothra 3. Kanva Gothra  & 4. Kātyāyana Gothra.
Most of the  details  are compilation from Wikipedia.

1. Dhanwantari Gothra

Dhanvantari  is an avatar of Vishnu from the Hindu tradition. He appears in the Vedas and Puranas as the physician of the Gods (devas), and the God of Ayurvedic medicine. It is common practice in Hinduism for worshipers to pray to Lord Dhanvantari seeking his blessings for sound health for themselves and/or others.
Dhanvantari was an early Indian medical practitioner and one of the world’s first surgeons. Based on Hindu traditions, he is regarded as the source of Ayurveda. He perfected many herbal based cures and natural remedies and was credited with the discovery of the antiseptic properties of turmeric and the preservative properties of salt which he incorporated in his cures.
Being a very skilled surgeon according to the standards of his time, he is widely believed to be the pioneer of modern medical practices like plastic surgery [1].
Albeit his methods were a lot cruder and more painful and were used only in emergencies, such as on the injuries of war victims.
All his surgeries were performed without anesthetic, however in spite of his crude methods he was reported to have had a very high success rate. As a result of the brilliance and achievements he displayed in the field of medicine he was chosen as one of the Nine Gems in early Indian ruler Vikramaditya’s court.
According to traditions, he taught surgery methods and procedures to Susrutha, the Father of Ayurvedic Surgeon.
The Legend
Dhanvantari is depicted as Vishnu with four hands, holding medical herbs in one hand and a pot containing rejuvenating nectar called amrita in another. The Puranas state that Dhanavantari emerged from the 'Ocean of Milk'.
Birthday celebration
Birth day celebration of Lord Dhanvantari, the God of health, healing and cure, is celebrated with great enthuiasm and happy environment, by the practitioners of the Ayurveda every year, on Dhan Teras, two days before Deepwali, the Hindu festival of Lights. In the Samudra Manthan, Lord Dhanvantari appeared with the keeping Amrit Pot, Shankha, Chakrra and Jalauka each one in his four hands.
Temples in India
In Northern India no permanent temple is established for Lord Dhanvantari. The reason is not yet known, but in Varanasey Sanssakrit Vishvavidyalaya, Varanasi, Uttar Pradesh state, one statue of Lord Dhanvantari is present in the Museum of the University.
However there are few dedicated temples to the Lord Dhanwantri, in South India especially in Kerala and Tamil Nadu, where Ayurvedic medicine is highly practised and patronised.
In Tamil Nadu, in the courtyard of Sri Ranganathaswamy Temple (Srirangam), is the Dhanvantari Temple where daily worshipping of the deity is performed. In the front of this temple there is an engraved stone believed to date around the 12th Century. The writing on the stone contains the details that Garud Vahan Bhattar, who was a great ayurvedic physician , established the statue inside the temple. As a 'Prasad' or 'Teerth', a decoction of the herbs is given to the visitors.
Though not known to many, there is a temple dedicated to Lord Dhanvantari in Kerala. It is in a village called Nelluvaya, 20 kilometers from guruvayur and trissur, located exactly midway between the two towns. The temple is believed to be as old as the temple of Guruvayur. Many ayurvedic doctors from kerala visit this temple before they start practicing medicine.
About 10 km from the railway station is the ages old, very powerful Dhanvantari Temple, Kannur (Cannanore), Kerala.
A not so well known temple in its humble environs has a serene atmosphere. Dhanvantari pooja is performed here for the good health of anyone, of any faith, from any corner of the world. Added attraction is the Huge Temple pond with water lilies!
Similarly, there's a Dhanvantri Kshetram (i.e.temple) in the outskirts of Calicut, in Kerala. This temple is gaining prominence, as people come from far off places to offer their prayers to the Lord, to cure them of their ailing diseases, or to be blessed with a healthy life ahead.
There is an exclusive very big temple for Lord Dhanwantari in Cherthala Maruthorvattom village in the Alleppy district. I have visited and made prayers in the temple.
In All India Ayurvedic practitioners worship Lord Dhanavantary.
Ashta Vaidya of Kerala
In Kerala, the family of "Ashta Vaidya" is famous and traditionally provide Ayurvedic and Siddha treatment to the sick. The forefathers of these Asta vaidyas are still today serving in the same manner as centuries ago. This family worships Lord Dhanvantari. Some family members have built temples inside their houses while others have built proper temples in his honour.
Near Kotakkalat Pulamantol village, here is a family of Ashta Vaidya. This family has a temple of Lord Dhanvantari. Vaidya Madam is near Vadakkancheri. Here the Ashta Vaidya Matra dattan have a statue of Dhanvantri, made of a mixture of five metals. In trishura's Perungva, a big temple is here built by Ashta vaidya. The Ashta Vaidya families are in the following places:
Cannanore (Kannur)

It seems that tradition of Lord Dhanwantri worshipping is regularly persisting in the families to families in Kerala.
While all the ashtavaidayan families (They are all Pushpaka Brahmins.) worship Dhanwantari it is felt they may be of Dhanwantari Gothra. Learnt members may wirite to me who belong to Pushpaka Brahmins group.
Dhanavantari - the God of Ayurveda
Kalyan Hindi monthly magazine, March 2001 issue, Geeta Press, Gorakhpur, UP
Source of References:
Lord Dhanvantari
Hindu God Dhanwantari: The promulgator of Ayurveda.
Does Ayurveda begin with Dhanvantari, the ancient physician? By D.P. Agrawal Dhanvantari in the Bhagavata Purana.

2. JamadagniGothra
Jamadagni is one of the Saptarishis (Seven Great Sages Rishi) in the seventh, i.e. the present Manvantara [1]. He is a descendant of the sage Bhrigu, one of the Prajapatis cretaed by Brahma, the God of Creation.
Jamadagni had several children with wife Renuka, the youngest of whom was Parashurama, an Avatara of Lord Vishnu.
Execution of Renuka
Renuka was such very devoted wife and the power of her chastity was manifest. Such was this power, that she used to fetch water from the river in a pot made of unbaked clay every day. The pot would hold together because of her devotion to her husband.
One day, when she was at the river, a handsome Gandharva happened to be passing by in the sky, in his chariot. Smitten with desire for this handsome youth, for merely an instant, the damage to her powers was done. The unbaked pot that she was carrying, dissolved into the river. She was no longer chaste of mind. Afraid to go back to her husband, she waited at the river bank.
Meanwhile Jamadagni, who was waiting for fresh water to begin his morning sacrifices, noticed that his wife had not yet returned from the river. By his yogic powers, he divined all that had taken place.
Exceedingly angry with his wife, he called his eldest son, told him what had happened and asked him to execute his mother. Horror-stricken, his son refused to perform this deed. He then asked all of his sons, in the order of their seniority, to execute their mother. While all the elder sons refused (and so Sage Jamadagni turned them to stone), only his youngest son, Parashurama, ever-obedient and righteous, at once beheaded his mother with his axe.
Jamadagni, pleased, offered to grant two boons to Parashurama, who at once asked that his mother be restored to life and his brothers to be unturned from stone and accepted into the family again. Impressed by his son's devotion and affection for his family, Jamadagni granted this boon and many others.
Jamadagni was later killed by a Kshatriya king Kartavirya Arjuna, over a dispute over a divine calf named kamadhenu.
Inhabitants of the Worlds Mahanirvana Tantra, translated by Arthur Avalon, (Sir John Woodroffe), 1913, Introduction and Preface.
3. Kanva Gothra.
Kanva is a renowned rishi, author of several hymns of the Rigveda, called a son of Ghora and one of the Angirasas. The Kanvas are the descendants of Kanva. Kanva is also the name of a founder of a Vedic shakha, of several princes and founders of dynasties and several authors. The Kanvas are also a class of evil spirits, against whom hymn 2.25 of the Atharvaveda is used as a charm.
Sahunthala  daughter of Viswamithra and Menaka was brought up by sage Kanva.The child born to Shakunthala is Bharat from whom our country got the name so.
4. Kātyāyana Gothra.
Kātyāyana (c. 3rd century BC) was a Sanskrit grammarian, mathematician and Vedic priest who lived in ancient India.
He is known for two works: The Varttika, an elaboration on Pānini grammar. Along with the Mahābhāsya of Patañjali, this text became a core part of the vyākarana (grammar) canon. This was one of the six Vedangas, and constituted compulsory education for Brahmin students in the following twelve centuries.
He also composed one of the later Sulba Sutras, a series of nine texts on the geometry of altar constructions, dealing with rectangles, right-sided triangles, rhombuses, etc.
Kātyāyana's views on the word-meaning connection tended towards naturalism. Kātyāyana believed, like Plato, that the word-meaning relationship was not a result of human convention. For Kātyāyana, word-meaning relations were siddha, given to us, eternal. Though the object a word is referring to is non-eternal, the substance of its meaning, like a lump of gold used to make different ornaments, remains undestroyed, and is therefore permanent.
This view may have been the nucleus of the Sphora doctrine enunciated by Bhartu hari in the 5th c., in which he elaborates the word-universal as the superposition of two structures — the meaning-universal or the semantic structure (artha-jāti) is superposed on the sound-universal or the phonological structure (śabda-jāti).
In the tradition of scholars like Pingala, Kātyāyana was also interested in mathematics. Here his text on the sulvasutras dealt with geometry, and extended the treatment of the Pythagorean theorem as first presented in 800 BC by Baudhayana.
Kātyāyana belonged to the Aindra School of grammarians and may have lived towards the North west of the Indian subcontinent.