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We consider first order symmetry operators for the equations of motion ofdifferential $p$-form fields in general $D$-dimensional background geometry ofany signature for both massless and massive cases. For $p=1$ and $p=2$ we givethe general forms of the symmetry operators. Then we find a class of symmetryoperators for arbitrary $p$ and $D$, which is naturally suggested by the lower$p$ results.

On First Order Symmetry Operators for the Field Equations of Differential Forms