Floating Bodies of Equilibrium in Three Dimensions. The central symmetric case

Three-dimensional central symmetric bodies different from spheres that can float in all orientations are considered. For relative density rho=1/2 there are solutions, if holes in the body are allowed. For rho different from 1/2 the body is deformed from a sphere. A set of nonlinear shape-equations determines the shape in lowest order in the deformation. It is shown that a large number of solutions exists. An expansion scheme is given, which allows a formal expansion in the deformation to arbitrary order under the assumption that apart from x=0,+1,-1 there is no x, which obeys P_{p,2}(x)=0 for two different integer ps, where P are Legendre functions.