  We consider a fourth order partial differential equation in n-dimensional
space introduced by Abreu in the context of K\"{a}hler metrics on toric
orbifolds. Similarity solutions depending only on the radial coordinate in R^n
are determined in terms of a second order ordinary differential equation. A
local asymptotic analysis of solutions in the neighbourhood of singular points
is carried out. The integrability (or otherwise) of Abreu's equation is
discussed.
