  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.
