Madonna's kids are duel nationals, so a bit different.
However, the answer to your question is no, the government do not make any provision for this. There is a six month visit visa, or a multiple entry visit visa (which will only allow him to stay for 6 months in any 12 months). There is no way he can get a settlement visa, as he is not going to settle.
The alternative would be for you to bring him here for the majority of the time, and get a settlement visa, but this is very, very difficult, as you would need to show that you have sole parental responsibility.
Vicky