  Since the expense of the numerical integration of large scale dynamical
systems is often computationally prohibitive, model reduction methods, which
approximate such systems by simpler and much lower order ones, are often
employed to reduce the computational effort. In this paper, for dynamical
systems with a first integral, new structure-preserving model reduction
approaches are presented that yield reduced-order systems while preserving the
first integral. We apply energy-preserving integrators to the reduced-order
systems and show some numerical experiments that demonstrate the favourable
behaviour of the proposed approaches.
