  We consider four (real or complex) dimensional hyper-K\"ahler metrics with a
conformal symmetry K. The three-dimensional space of orbits of K is shown to
have an Einstein-Weyl structure which admits a shear-free geodesics congruence
for which the twist is a constant multiple of the divergence. In this case the
Einstein-Weyl equations reduce down to a single second order PDE for one
function. The Lax representation, Lie point symmetries, hidden symmetries and
the recursion operator associated with this PDE are found, and some group
invariant solutions are considered.
