Every underdetermined system of partial differential equations arising from a variational principle admits an infinite hierarchy of higher-order generalized symmetries. These symmetries are a consequence of the Noether dependencies among the Euler–Lagrange equations that follow from Noether's Second Theorem. This result is a consequence of a more general theorem on the existence of higher-order generalized symmetries for any system of differential equations that admits an infinitesimal symmetry generator depending on an arbitrary function of the independent variables.

中文翻译：

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偏微分方程和诺特第二定理的欠定系统的高阶对称性

每个由变分原理产生的偏微分方程的欠定系统都承认高阶广义对称性的无限层次结构。这些对称性是遵循诺特第二定理的欧拉-拉格朗日方程之间的诺特相关性的结果。这个结果是关于任何微分方程系统存在高阶广义对称性的更一般定理的结果，该系统允许依赖于自变量的任意函数的无穷小对称生成器。