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A formula for symmetry recursion operators from non-variational symmetries of partial differential equations

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.