This technical note studies some properties of the recently developed parametric Lyapunov equation based low gain feedback design method. As applications of these new properties, alternative and simpler solutions are proposed to the (global) stabilization problem for a class of linear systems with input delay and the semi-global stabilization problem when the systems are in addition subject to actuator saturation. Besides the simplicity in their construction, the new solutions can also be easily scheduled online to achieve global result in the presence of input saturation.