  We study local, analytic solutions for a class of initial value problems for
singular ODEs. We prove existence and uniqueness of such solutions under a
certain non-resonance condition. Our proof translates the singular initial
value problem to an equilibrium problem of a regular ODE. Then, we apply
classical invariant manifold theory. We demonstrate that the class of ODEs
under consideration captures models which describe the shape of axially
symmetric surfaces which are closed on one side. Our main result guarantees
smoothness at the tip of the surface.
