In spinor formalism, since any massless free-field spinor with spin higher
    than $1/2$ can be constructed with spin-1/2 spinors (Dirac-Weyl spinors) and
    scalars, we introduce a map between Weyl fields and Dirac-Weyl fields. We
    determine the corresponding Dirac-Weyl spinors in a given empty spacetime.
    Regarding them as basic units, other higher spin massless free-field spinors
    are then identified. Along this way, we find some hidden fundamental features
    related to these fields. In particular, for non-twisting vacuum Petrov type N
    solutions, we show that all higher spin massless free-field spinors can be
    constructed with one type of Dirac-Weyl spinor and the zeroth copy.
    Furthermore, we systematically rebuild the Weyl double copy for non-twisting
    vacuum type N and vacuum type D solutions. Moreover, we show that the zeroth
    copy not only connects the gravity fields with a single copy but also connects
    the degenerate Maxwell fields with the Dirac-Weyl fields in the curved
    spacetime, both for type N and type D cases. Besides, we extend the study to
    non-twisting vacuum type III solutions. We find a particular Dirac-Weyl scalar
    independent of the proposed map and whose square is proportional to the Weyl
    scalar. A degenerate Maxwell field and an auxiliary scalar field are then
    identified. Both of them play similar roles as the Weyl double copy. The result
    further inspires us that there is a deep connection between gravity theory and
    gauge theory.

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