O(n) invariant solutions of Abreu's equation

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.