  We prove that measure-preserving symmetries of an $n$-dimensional
differential system preserve its divergence and the divergence derivatives
along the solutions. Also, we prove that measure-preserving reversibilities
preserve odd-order divergence derivatives along the solutions, and that
even-order derivatives are multiplied by $-1$. We apply such results to find
all the area-preserving symmetries and reversibilities of planar Lotka-Volterra
and Li\'enard systems.
