This paper considers a conservative system admitting a family of single-frequency oscillations with a domain Ω. For the original system, an autonomous controlled (ε-corrected) system with a small gain is introduced; a given oscillation from the domain Ω is stabilized by constructing a cycle that attracts all trajectories from this domain together with its ε- neighborhood. A universal adaptive control law, acting as a nonlinear force linear in velocity, is designed to track the current value of potential energy during motion. The cycle is constructed for any system oscillation. As a result, a new class of autonomous controlled systems is obtained based on the conservative system, and the operating modes of this class are stabilized (in the large) cycles with any desired energy. Examples are provided.