  In this paper, we study symmetric integrators for solving second-order
ordinary differential equations on the basis of the notion of continuous-stage
Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on
the Legendre expansion technique in conjunction with the symmetric conditions
and simplifying assumptions for order conditions. New families of symmetric
integrators as illustrative examples are presented. For comparing the numerical
behaviors of the presented methods, some numerical experiments are also
reported.
