  We give new simple direct proofs in all spacetimes for the existence of
asymmetric $(n,m+1)$-spinor potentials for completely symmetric
$(n+1,m)$-spinors and for the existence of symmetric $(n,1)$-spinor potentials
for symmetric $(n+1,0)$-spinors. These proofs introduce a `superpotential',
i.e., a potential of the potential, which also enables us to get explicit
statements of the gauge freedom of the original potentials. The main
application for these results is the Lanczos potential $L_{ABCA'}$, of the Weyl
spinor and the electromagnetic vector potential $A_{AA'}$. We also investigate
the possibility of existence of a {\em symmetric} potential $H_{ABA'B'}$ for
the Lanczos potential, and prove that in {\em all Einstein spacetimes} any
symmetric (3,1)-spinor $L_{ABCA'}$ possesses a symmetric potential
$H_{ABA'B'}$. Potentials of this type have been found earlier in investigations
of some very special spinors in restricted classes of spacetimes. All of the
new spinor results are translated into tensor notation, and where possible
given also for four dimensional spaces of arbitrary signature.
