  The primary objective of this paper is to present a long-term numerical
energy-preserving analysis of one-stage explicit symmetric and/or symplectic
extended Runge--Kutta--Nystr\"{o}m (ERKN) integrators for highly oscillatory
Hamiltonian systems. We study the long-time numerical energy conservation not
only for symmetric integrators but also for symplectic integrators. In the
analysis, we neither assume symplecticity for symmetric methods, nor assume
symmetry for symplectic methods. It turns out that these both kinds of ERKN
integrators have a near conservation of the total and oscillatory energy over a
long term. To prove the result for symmetric integrators, a relationship
between symmetric ERKN integrators and trigonometric integrators is established
by using Strang splitting and based on this connection, the long-time
conservation is derived. For the long-term analysis of symplectic ERKN
integrators, the above approach does not work anymore and we use the technology
of modulated Fourier expansion developed in SIAM J. Numer. Anal. 38 (2000) by
Hairer and Lubich. By taking some novel adaptations of this essential
technology for non-symmetric methods, we derive the modulated Fourier expansion
for symplectic ERKN integrators. Moreover, it is shown that the symplectic ERKN
integrators have two almost-invariants and then the near energy conservation
over a long term is obtained.
