An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a general formula that produces a pre-symplectic operator from a non-gradient adjoint-symmetry. These formulas are illustrated by several examples of linear PDEs and integrable nonlinear PDEs. Additionally, a classification of quasilinear second-order PDEs admitting a multiplicative symmetry recursion operator through the first formula is presented.

中文翻译：

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偏微分方程非变对称性的对称递推算子公式

从连接变分积分因子和非变分对称性的新结果中，可以找到一个明确的公式来找到偏微分方程（PDE）的对称递归运算符。该公式是通用公式的特例，该通用公式通过非梯度伴随对称产生一个先符号运算符。这些公式由线性PDE和可积分非线性PDE的几个示例说明。此外，提出了准线性二阶PDE的分类，该二阶PDE通过第一公式允许乘性对称递归算子。