  In this paper, combining the ideas of exponential integrators and discrete
gradients, we propose and analyze a new structure-preserving exponential scheme
for the conservative or dissipative system $\dot{y} = Q(M y + \nabla U (y))$,
where $Q$ is a $d\times d$ skew-symmetric or negative semidefinite real matrix,
$M$ is a $d\times d$ symmetric real matrix, and $U :
\mathbb{R}^d\rightarrow\mathbb{R}$ is a differentiable function. We present two
properties of the new scheme. The paper is accompanied by numerical results
that demonstrate the remarkable superiority of our new scheme in comparison
with other structure-preserving schemes in the scientific literature.
