  As is known that various dynamical systems including all Hamiltonian systems
preserve volume in phase space. This qualitative geometrical property of the
analytical solution should be respected in the sense of Geometric Integration.
This paper analyses the volume-preserving property of exponential integrators
in different vector fields. We derive a necessary and sufficient condition of
volume preservation for exponential integrators, and with this condition,
volume-preserving exponential integrators are analysed in detail for four kinds
of vector fields. It turns out that symplectic exponential integrators can be
volume preserving for a much larger class of vector fields than Hamiltonian
systems. On the basis of the analysis, novel volume-preserving exponential
integrators are derived for solving highly oscillatory second-order systems and
extended Runge--Kutta--Nystr\"{o}m (ERKN) integrators of volume preservation
are presented for separable partitioned systems. Moreover, the volume
preservation of Runge--Kutta--Nystr\"{o}m (RKN) methods is also discussed. Four
illustrative numerical experiments are carried out to demonstrate the notable
superiority of volume-preserving exponential integrators in comparison with
volume-preserving Runge-Kutta methods.
