We consider a fourth order nonlinear partial differential equation
in n-dimensional space introduced by Abreu in the context of Kahler
metrics on toric varieties. Rotation invariant similarity solutions,
depending only on the radial coordinate in Rn, are
determined from the solutions of a second order ordinary differential
equation (ODE), with a non-autonomous Lagrangian formulation. A local
asymptotic analysis of solutions of the ODE in the neighbourhood of
singular points is carried out, and the existence of a class of
solutions on an interval of the positive real semi-axis is proved
using a nonlinear integral equation. The integrability (or otherwise)
of Abreu's equation is discussed.