  A twistor construction of the hierarchy associated with the hyper-K\"ahler
equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in
four dimensions) is given. The recursion operator R is constructed and used to
build an infinite-dimensional symmetry algebra and in particular higher flows
for the hyper-K\"ahler equations. It is shown that R acts on the twistor data
by multiplication with a rational function. The structures are illustrated by
the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended
space-time ${\cal N}$ is constructed whose extra dimensions correspond to
higher flows of the hierarchy. It is shown that ${\cal N}$ is a moduli space of
rational curves with normal bundle ${\cal O}(n)\oplus{\cal O}(n)$ in twistor
space and is canonically equipped with a Lax distribution for ASDVE
hierarchies. The space ${\cal N}$ is shown to be foliated by four dimensional
hyper-K{\"a}hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian
formulations of the ASDVE in the form of the heavenly equations are given. The
symplectic form on the moduli space of solutions to heavenly equations is
derived, and is shown to be compatible with the recursion operator.
