By using the self-dual Yang-Mills (SDYM) equation as an example, we study a method for relatingsymmetries and recursion operators of two partial differential equations connected to each other by anon-auto-Backlund transformation. We prove the Lie-algebra isomorphism between the symmetriesof the SDYM equation and those of the potential SDYM (PSDYM) equation, and we describe theconstruction of the recursion operators for these two systems. Using certain known aspects ofthe PSDYM symmetry algebra, we draw conclusions regarding the Lie algebraic structure of the"potential symmetries" of the SDYM equation.
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