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This paper deals with Dirac, twistor and Killing equations on Weyl manifolds with $C$-spin structures. A conformal Schr\"odinger-Lichnerowicz formula is presented and used to derive integrability conditions for these equations. It is shown that the only non-closed Weyl manifolds of dimension greater than 3 that admit solutions of the real Killing equation are 4-dimensional and non-compact. Any Weyl manifold of dimension greater than 3, that admits a real Killing spinor has to be Einstein-Weyl.
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