Most of us are aware of the children of Shiva and Parvati ⇒ Kartikeya, Ganesh and many others.
I have never come across the children of Vishnu and Lakshmi.
Lastly, I am aware of the avatars of Vishnu and Lakshmi who had children, e.g.
Who are the children of Vishnu and Lakshmi? Can anyone provide me with this knowledge?
As far as I know, the only time that children of Lakshmi are mentioned in Hindu scriptures is in verses 11 - 12 of the Sri Suktam, a hymn from the Khilani (apocrypha or later portions) of the Rig Veda:
We are the progeny of our forefather, Sage Kardama, who is one of the sons of Goddess Lakshmi. We invoke that Sage Kardama to install in his family the Universal Mother, Mahalakshmi, who is decked with the garland of lotuses. So be it. We invoke another son of Lakshmi named Chikleeta. May he dwell in our home and may his mother, Mahalakshmi, dwell in our family.
There's almost no information known beyond that. The sage Kardama is usually thought of as a son of Brahma, born from Brahma's shadow, as mentioned in this verse of the Srimad Bhagavatam. The only other connection between Lakshmi and Kardama is a story about how when Lakshmi emerged out of the churning of the ocean, she took sage Kardama as her adopted father, not son, as a reward for his Tapasya. And I don't think Chiklita is mentioned in any other scriptures.
The meanings of the names Kardama and Chiklita both refer to moist or wet things, so that might also be related to Lakshmi's emergence from the churning of the ocean,
One more thing that you may want to know is that Kardama and Chilkita are apparently two of the seers of the verses in the Sri Suktam, according to this webpage:
If we consider the entire Sukta as a whole, then the Rishis are: Each mantra of this Sukta has a different Rishi (seer), Chandas (metre), Devi (deity) and Viniyoga (ritual application). Ananda, Chikleeta, Kardama, Shreeda and Indira, who are said to be the sons of goddess Lakshmi.
But I'm not sure whether the claim that these are all sons of Lakshmi has any scriptural basis.
EDIT: This excerpt from the Vayu Purana lists two named more sons of Vishnu and Lakshmi, "Bala (Strength) and Utsaha (Energy)". And it also says the drivers of vimanas were born to her as "mental sons", and that she had two daughters, Ayati and Niyati, who became daughters-in-law of Brighu (who is Lakshmi's adopted father!).
As per the BrahmAnda PurAna, Lord Vishnu and Lakshmi Devi have two sons called Bala (strength) and UtsAha (enthusiasm, energy).
The relevant verses are given below:
Bhrigoh khyAtirvijajnehatha iswarau sukhadukkhayoh | SubhAsubhapradAtarau sarva prAna VrtAviha || Devau dhAtA vidhAtArau manvantara vichArinau | TayorjyeshthA tu bhagini devi srir loka bhAvini || SA tu nArAyanam devam pati mAsAdya shobhanam | NArAyanatmajau sAddvi valotsAhou vyajAyat ||
...........
Bhrigu and his wife KhyAti gave birth to two sons named DhAtA and VidhAta and a daughter named Sri or Lakshmi. Lakshmi obtained as her husband NArAyana and thereafter gave birth to two sons called Bala (strength) and UtsAha (enthusiasm).
BrahmAnda PurAna 29| 1-3.
They are a few children that I have found.They are:
1)Ekvira.You can find his full story here in Keshav Srinivasan's answer here: Why Revantha the youngest Son of Sun god went to Vishnu's abode? In short,Lakshmi incarnated as a horse and she and Vishnu had a son,Ekvira,who is adopted by a mortal king.Ekvira evantually marries a princess,whom he saves from a ogre.
2)Amritavalli and Sundravalli.They
born out of the tears of joy of Narayana during his incarnation as Trivikrama
When Lord Subrahmanya was staying at Kanda Verpu, the two daughters of Lord Mahavishnu, Amritavalli and Sundaravalli, who cherished the desire of becoming the consorts of Subrahmanya, went to Sanavana Poigai and commenced austere penance to fulfil their desires. Pleased with their prayer and worship, Lord Subrahmanya appeared before them and told Amritavalli: "You will be brought up by Indra as his daughter and I shall marry you in due course.
"Her younger sister Sundaravalli was also graced with a similar blessing. She was born to Sage Sivamuni and brought up by Nambi, the headman of hunters. Amritavalli took the form of a female child and went to Mount Meru the abode of Indra, and told him: "I am the daughter of Mahavishnu and the responsibility of looking after me has been entrusted to you.' On hearing this, Indra became very happy and directed Airavatam, his white elephant, to take care of the child.
sources:http://murugan.org/temples/tiruparankundram-legends.htm and http://tamildeities.blogspot.sg/2013/08/amirtavalli-sundaravalli.html?m=1
3)Kamadeva.Kama,or Manmatha,is said to be the son of both Vishnu and Lakshmi.Furthermore,he incarnated as Prayudma,the son of Krishna and Rukmini.
4)The eighteen sons of Lakshmi.It is believed that Devi Lakshmi herself is blessed with eighteen sons. The devotees chant following names on every Friday to please and to get blessings of Goddess Lakshmi.You can see the list here: http://www.drikpanchang.com/hindu-goddesses/lakshmi/lakshmi-children/goddess-lakshmi-sons-names.html
I came to know of some sons of Lord Vishnu but they were not with Goddess Lakshmi but with some other women. According to Chapter 22, verse 47 of ShataRudraSamhita in the Shiva Purana, Lord Vishnu had begot sons with some damsels (apsaras) born after the Samudra Manthan. These sons created a lot of havoc on earth and in heaven causing misery to all, therefore on request of Gods, Lord Shiva later killed them all by taking an incarnation of a bull. Here is the extract from the ShataRudraSamhita:
Chapter 7, Pūrvārdha, Kāśī-khaṇḍa
महाकालमहाकालमहाकालेतिसंततम् ।
स्मरतःस्मरतो नित्यं स्मरकर्तृस्मरांतकौ ॥ ९९ ॥
- The Father of Smara (Viṣṇu) and the Slayer of Smara (Śiva) always remember those who remember and repeat the name Mahākāla constantly.
English Translation by G.V. Tagare
Smara (स्मर) is another name of 'the god of Love' - Kāmadeva
