This is an idiom. It means to manage the situation. Just to talk about rule we place `noun / pronoun` in place of `to` and `ones` and use it in various tenses.like... I keep my head above water most of the times..means I manage the situation well most of the times.He keeps his head above water.She keeps her head above water.They keep their heads above water.My son keeps his head above water.That day I kept my head above water.He could not keep his head above water.You should keep your head above water.Could you keep your head above water? etc.This also means maintaining cool head while in the interaction or handling a situation.Enjoy !