jaime and cersei are also aerys kids. he was infatuated with joanna lannister and he raped her. the reason why ilyn payne had his tongue cut out was because he witnessed the event. and it is also also why he laughs when jaime admits his love for cersei to him.
tywin's only son, and heir, is tyrion. i don't know why it's not alluded to more in the show. i think maybe everyone is obsessed with wanting to know who jon snow's mum is.