This paper examines the pathological behavior of the invariant set of bounded orbits for the differential equation f = -(%)’ where z = x1 + ha is a complex variable. The dots denote time differentiation and the bar denotes complex conjugation. Let x = (X 1 , x2), y = (yl , y2), be vectors in the plane, ( , ) the usual planar Euclidean inner product, and 11 * :I = ( , )l12. Then this differential equation is equivalent to the analytic Hamiltonian system of two degrees of freedom