  In \cite{B3}, Bridgeland defined a geometric structure, named a Joyce
structure, conjectured to exist on the space $M$ of stability conditions of a
$CY_3$ triangulated category. A feature of this structure is a complex
hyper-K\"ahler metric with homothetic symmetry on the total space $X = TM$ of
the holomorphic tangent bundle. \par Generalising the isomonodromy calculation
which leads to the $A_2$ Joyce structure in \cite{BM}, we obtain an explicit
expression for a hyper-K\"ahler metric with homothetic symmetry via
construction of the isomonodromic flows of a Schr\"odinger equation with
deformed polynomial oscillator potential of odd degree $2n+1$. The metric is
defined on a total space $X$ of complex dimension $4n$ and fibres over a
$2n$--dimensional manifold $M$ which can be identified with the unfolding of
the $A_{2n}$-singularity. The hyper-K\"ahler structure is shown to be
compatible with the natural symplectic structure on $M$ in the sense of
admitting an \textit{affine symplectic fibration} as defined in \cite{BS}. \par
Separately, using the additional conditions imposed by a Joyce structure, we
consider reductions of Pleba\'nski's heavenly equations that govern the
hyper-K\"ahler condition. We introduce the notion of a \textit{projectable
hyper-Lagrangian} foliation and show that in dimension four such a foliation of
$X$ leads to a linearisation of the heavenly equation.
