Abstract
Deals with the following question: given a symmetry vector field Y of a system of second-order ordinary differential equations, and an associated constant of the motion F, is it possible to find a Lagrangian L for the system, such that Y becomes a Noether symmetry with respect to L, and F its implied Noether constant? For one degree of freedom systems the answer to this question is affirmative. In addition, attention is paid to the construction of a suitable constant of the motion F for given symmetry Y and vice versa. Several examples are discussed.