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 Liénard systems.

中文翻译：

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普通微分系统的保度量对称性和可逆性

我们证明了n维微分系统的保度量对称性保持了其散度和沿着解的散度导数。此外，我们证明了保持度量的可逆性沿解保留了奇数阶散度导数，并且偶数阶导数乘以-1。我们应用这些结果来发现平面Lotka-Volterra和Liénard系统的所有保留面积对称性和可逆性。