I got to know from a relative that Shani has 2 sons - Maandhi & Kuligan.
Just like Shani, they are supposed to create malefic effects according to Jyotisha. What is the story behind them? Very little seems to be known about them.
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Maandhi & Kuligan as son's of Shani seem to be only known among Tamil Hindus. Some say Maandhi & Kuligan are 2 names for the same son of Shani and some say they are 2 different sons.
They are seen along with Shani and his 2 wives in Ramanathaswami Temple at Thirunaraiyur (Nachiarkoil) near Kumbakonam.
In the book Sarvartha Sindhamani, it is mentioned that Maandhi is Shani's son and Kuligan is Yama's son.
There is a Tamil folklore from Ramayan about Maandhi's birth: When Raavana's son Indrajit was about to be born, Shani was poised to be in the 12th house of his horoscope which is a malefic house. Raavana forced Shani to be in the 11th house which is a beneficial house instead. Shani was forced to do as per Raavan's wishes but he had kept one of his legs partially in the 12th house. On seeing this, angry Raavana took his sword and chopped Shani's leg. The chopped portion fell in the 1st house and Maadhi arose from it and gave malefic effects to Indrajit(by giving him short lifespan).
The story surrounding Shani's sons seem to be associated only with Jyotisha, endemic to South India and not much known outside that circle. Maandhi is supposed to be a shadow planet bringing ill effects.
Shani with his wives and 2 sons from Ramanathaswami Temple, Thirunaraiyur :
 http://classroom2007.blogspot.com/2013/07/astrology_24.html (Text in Tamil)
 http://bit.ly/1rNZzFF (Text in Tamil)