Our study relates to systems whose dynamics generalize x = h(y, u), y = f(y, u), where the state components x integrate functions of the other components y and the inputs u. We give sufficient conditions under which global asymptotic stabilizability of the y subsystem (respectively, by saturated control) implies global asymptotic stabilizability of the overall system (respectively, by saturated control), It is obtained by constructing explicitly a control Lyapunov function and provides feedback laws with several degrees of freedom which can be exploited to tackle design constraints, Also, we study how appropriate changes of coordinates allow us to extend its domain of application, Finally we show how the proposed approach serves as a basic tool to be used, in a recursive design, to deal with more complex systems, In particular the stabilization problem of the so-called feedforward systems is solved this way.