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We briefly review the hierarchy for the hyper-K\"ahler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-K\"ahler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is tri-holomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a `twistor group' action as introduced by Bielawski \cite{B00}. This leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalised Legendre transform for the hyper-K\"ahler metric itself given by Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of hyper-K\"ahler metrics admitting three tri-holomorphic Killing spinors. These metrics are in this sense analogous to the 'finite gap' solutions in soliton theory. Finally we extend the concept of a hierarchy from that of \cite{DM00} for the four-dimensional hyper-K\"ahler equations to a generalisation of the conformal anti-self-duality equations and briefly discuss hidden symmetries for these equations., Comment: Final version. To appear in the August 2003 special issue of JMP on `Integrability, Topological Solitons, and Beyond'

Since in all the type-D solutions of the Einstein vacuum field equations with a cosmological term each maximal spin-weighted component of the electromagnetic spinor field satisfies a decoupled equation and there exists a two-index Killing spinor field; we show, via the adjoint operators method, that a symmetry operator for the Maxwell equations can be constructed.

In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin_n of the Riemannian holonomy group, with fixed points on the spin representation space. In particular, we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors., 10 pages, LaTeX2e

It is shown that in a type {2,2} solution of the Einstein vacuum field equations, which admits a two‐index Killing spinor, a differential operator can be constructed that maps a solution of the Rarita–Schwinger equation into a solution of its complex conjugate. Furthermore, by considering the Plebanski–Demianski metric, which contains all the vacuum type {2,2} metrics, it is shown that the separable solutions of the Rarita–Schwinger equation are eigenfunctions of a certain differential operator with the Starobinsky constant as the eigenvalue.

The concept of Killing spinor is analyzed in a general way by using the spinorial formalism. It is shown, among other things, that higher derivatives of Killing spinors can be expressed in terms of lower order derivatives. Conformal Killing vectors are studied in some detail in the light of spinorial analysis: Classical results are formulated in terms of spinors. A theorem on Lie derivatives of Debever–Penrose vectors is proved, and it is shown that conformal motion in vacuum with zero cosmological constant must be homothetic, unless the conformal tensor vanishes or is of type N. Our results are valid for either real or complex space–time manifolds.

It is shown that in a type‐D space‐time that admits a two‐index Killing spinor a differential operator can be constructed that maps a solution of the Maxwell equations into another solution. By considering as a background the Plebanski–Demianski metric, which includes all the vacuum type‐D metrics, this operator is used to obtain all the components of the electromagnetic field and the vector potential. The separated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigensolutions of a certain differential operator with the ‘‘Starobinsky constant’’ as the eigenvalue.

Type D metrics in vacuum or with a special electromagnetic field are studied using the fact that they admit a D (1,0) Killing spinor. It is shown that the E and Φ potentials of Ernst can be obtained in a straightforward way for these types of metrics.

It is shown that in a vacuum space‐time, possibly with a nonzero cosmological constant, which admits a D(1,0) Killing spinor, one component of the perturbed Weyl spinor that satisfies a decoupled equation, when multiplied by an appropriate factor made out of the components of the Killing spinor, constitutes a Debye potential that generates metric perturbations of the considered background. It is also shown that in the case where the background is of type N, there is an operation that relates the gravitational perturbations and the zero‐rest‐mass fields of spin‐0, ‐ 1/2 , and ‐1.

It is shown that the separation constant not related to the space‐time symmetries, which appears in the solution of the source‐free Maxwell equations on a type‐D vacuum background with cosmological constant, can be defined in an invariant way as the eigenvalue of a differential operator made out of a two‐index Killing spinor, with the eigenfunctions being the separable solutions of the Maxwell equations.