  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.
