  This paper is concerned with the Floating Body Problem of S. Ulam: the
existence of objects other than the sphere, which can float in a liquid in any
orientation. Despite recent results of F. Wegner pointing towards an
affirmative answer, a full proof of their existence is still unavailable. For
objects with cylindrical symmetry and density 1/2, the conditions of neutral
floating are formulated as an initial value problem, for which a unique
solution is predicted in certain cases by a suitable generalization of the
Picard-Lindel\"of theorem. Numerical integration of the initial value problem
provides a rich variety of neutrally floating shapes.
