Isochronous sets of invariant control systems

Let G be a connected Lie group with Lie algebra g and Σ=(G,D) a controllable invariant control system. A subset A⊂G is said to be isochronous if there exists a uniform time ^TA>0 such that any two arbitrary elements in A can be connected by a positive orbit of Σ at exact time ^TA. In this paper, we search for classes of Lie groups G such that any Σ has the following property: there exists an increasing sequence of open neighborhoods (^Vn)n<0 of the identity in G such that the group can be decomposed in isochronous rings ^Wn=Vn+₁- ^Vn. We characterize this property in algebraic terms and we show that three classes of Lie groups satisfy this property: completely solvable simply connected Lie groups, semisimple Lie groups and reductive Lie groups.