How do you get the new Kairos Dragon in Dragonvale?

Solution 1:

I found some good information here:

Dragonvale Kairos Dragon

"You cannot breed or buy Kairos, you have to summon him! First you have to buy the Perch of Kairos (750,000 coins). Then you need to gather three artifacts: the Chronolith, the Time Claw and the Statue of Kairos. The artifacts are built by collecting 21 time fragments. They can be won on the Dragon Races or bought for 2,700 gems total."

Solution 2:

You can't buy, sell or breed the Kairos Dragon. You need to buy the perch of Kairos, which comes under habitats or buildings. There are three parts to it and seven pieces to each part. You must complete each of them before you can summon the dragon. You can get pieces by racing on the dragon track. As far as I know, you can only get one per day.

Solution 3:

The Karios Dragon in Dragonvale is not breedable, but you can earn it by winning Dragonvale Races apparently.

You could check out this guide about Dragonvale Breeding it might help you with others too

Dragonvale Breeding Guide