  This paper analyses the long-time behaviour of one-stage symplectic or
symmetric extended Runge--Kutta--Nystr\"{o}m (ERKN) methods when applied to
nonlinear wave equations. It is shown that energy, momentum, and all harmonic
actions are approximately preserved over a long time for one-stage explicit
symplectic or symmetric ERKN methods when applied to nonlinear wave equations
via spectral semi-discretisations. For the long-term analysis of symplectic or
symmetric ERKN methods, we derive a multi-frequency modulated Fourier expansion
of the ERKN method and show three almost-invariants of the modulation system.
In the analysis of this paper, we neither assume symmetry for symplectic
methods, nor assume symplecticity for symmetric methods. The results for
symplectic and symmetric methods are obtained as a byproduct of the above
analysis. We also give another proof by establishing a relationship between
symplectic and symmetric ERKN methods and trigonometric integrators which have
been researched for wave equations in the literature.
